2-CARTESIAN FIBRATIONS II: A GROTHENDIECK CONSTRUCTION FOR $\infty $-BICATEGORIES

Abstract

In this work, we conclude our study of fibred $\infty $-bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set S (which need not be fibrant) we construct a 2-categorical version of Lurie’s straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\infty $-bicategory of 2-Cartesian fibrations over S and the $\infty $-bicategory of contravariant functors with values in the $\infty $-bicategory of $\infty $-bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.

1. Introduction

This is the second of a two-paper sequence devoted to $(\infty ,2)$ -categorical fibrations and the concomitant Grothendieck constructions, with an eye towards understanding $(\infty ,2)$ -categorical cofinality. In this paper, we provide an explicit and computationally tractable Grothendieck construction for $(\infty ,2)$ -categories fibred in $(\infty ,2)$ -categories. In a companion paper [2], we leverage this technology to prove a complete characterisation of marked cofinal functors of $(\infty ,2)$ -categories, which we conjectured in [3].

The Grothendieck construction first emerged as a tool to study descent in [18], but it has since become an invaluable tool to study universal properties more generally. In its original form, it takes the form of an equivalence

for any small category , between the category of fibred categories over , and the category of pseudo-functors . The underlying idea is that certain conditions on a functor mean that the fibres of p vary (pseudo-)functorially in . Indeed, the original definition of a fibred category, in [17], was what we today would call a pseudo-functor . More precisely, an assignment of a category $F(x)\in \operatorname {Cat}$ for every , a functor $F(f):F(y)\to F(x)$ for every morphism $f:x\to y$ in and natural isomorphisms $F(g)\circ F(f)\cong F(f\circ g)$ for every composable pair of morphisms, satisfying additional coherence conditions.

The Grothendieck construction reformulates the data of a pseudo-functor into a Cartesian fibration. Given a functor , a morphism $f:x\to y$ in is called Cartesian if, for every $g:z\to y$ in , and every commutative diagram

in , there is a unique morphism $\tilde {h}:z\to x$ with $P(\tilde {h})=h$ , such that $f\circ \tilde {h}=g$ . The functor P is said to be a Cartesian fibration if, for every $f:c\to P(y)$ in , there is a Cartesian morphism $\tilde {f}:x\to y$ in such that $P(\tilde {f})=f$ .

The equivalence between pseudo-functors and Cartesian fibrations over is then achieved by constructing a Cartesian fibration as follows:

• • The objects of $\operatorname {El}(F)$ consist of pairs $(c,x)$ , where , and $x\in F(c)$ .

• • A morphism $(f,\tilde {f}):(c,x)\to (d,y)$ consists of a morphism $f:c\to d$ in , together with a morphism $\tilde {f}:x\to F(f)(y)$ in $F(x)$ .

The Cartesian morphisms of $\operatorname {El}(F)$ are precisely those $(f,\tilde {f})$ such that $\tilde {f}$ is an isomorphism.

1.1. Higher-categorical Grothendieck constructions

More recent incarnations of the Grothendieck construction have focused on $\infty $ -categorical variants. By their very nature, functors of $(\infty ,1)$ -categories generalise pseudo-functors of $(2,1)$ -categories,¹ so that higher Grothendieck constructions now take the form of equivalences

of $\infty $ -categories. This equivalence was proven by Lurie in [23], using model-categorical techniques which we adapt in the present work.

The basic form of these arguments is not hard to follow. Given an $\infty $ -category , presented as a quasi-category, Lurie defines marked simplicial sets over to be a pair $(X,M_X)$ consisting of a simplicial set $X\in {\operatorname {Set}}_\Delta $ and a subset $M_X\subset X_1$ of marked edges containing all degenerate edges, equipped with a morphism of simplicial sets. Requiring maps to preserve these marked edges yields a category . Lurie then constructs a model structure on this category, the fibrant objects of which satisfy lifting properties akin to those defining 1-categorical Cartesian fibrations. In particular, the corresponding model structure on ${{\operatorname {Set}}_{\Delta }^+}\cong ({{\operatorname {Set}}_{\Delta }^+})_{/\Delta ^0}$ models $(\infty ,1)$ -categories.

With these model structures in place, one can consider the category of simplicially-enriched functors , and equip it with the projective model structure. The $(\infty ,1)$ -categorical Grothendieck construction then takes the form of a Quillen equivalence

between these two model categories.

In the $\infty $ -categorical context, Grothendieck constructions have become an indispensable tool, as the added computational complexity of $\infty $ -categorical constructions renders many ad-hoc constructions of functors nearly impossible to work with. It is often far easier to work with the fibration associated to a functor of $\infty $ -categories than with the functor itself. Examples of such applications include the study of monoidal $(\infty ,1)$ -categories in [22] and [24] and the approach to lax colimits presented in [15]. The study of higher forms of cofinality, which is our intended application, is another case in which it is virtually essential to use the Grothendieck construction.

1.2. Scaled simplicial sets and non-fibrant base

Throughout this work, we model $(\infty ,2)$ -categories using scaled simplicial sets. Introduced by Lurie in [25], this model is similar in perspective to the marked simplicial sets described above. More formally, a scaled simplicial set is a pair $(X,T_X)$ consisting of a simplicial set $X\in {\operatorname {Set}}_{\Delta }$ , together with a subset $T_X\subset X_2$ containing all degenerate 2-simplices. We think of the simplices in $T_X$ – typically referred to as thin triangles or thin two simplices – as representing invertible 2-morphisms, and we think of all other two simplices as representing non-invertible 2-morphisms. There are two reasonable conventions for the direction of this 2-morphism, and we here take the convention that a 2-simplex is an ‘upward-pointing’ 2-morphism, i.e.

We briefly recapitulate how to view a strict 2-category from this perspective in section 2.1.

Our use of this model for $(\infty ,2)$ -categories presents a number of advantages. Most immediately apparent, much intuition from quasi-categorical models of $(\infty ,1)$ -categories carries over to scaled simplicial sets, and such intuitions suffuse all of our proofs. More importantly though, our model-categorical approach to the Grothendieck construction means that our proof holds even when the base of our fibrations is a non-fibrant scaled simplicial set.

The benefit of this generality may not be immediately apparent, but arises from the consideration of lax functors. A morphism of scaled simplicial sets $(X,T_X)\to (Y,T_Y)$ sends scaled 2-simplices to scaled 2-simplices, and thus sends invertible 2-morphisms to invertible 2-morphisms. In this sense, morphisms of scaled simplicial sets between fibrant objects generalise pseudofunctors of 2-categories. However, if one considers a fibrant scaled simplicial set $(X,T_X)$ (what we will later call an $\infty $ -bicategory), and replaces the scaling $T_X$ with the scaling $M_X$ which consists of thin triangles where the edge or the edge is an equivalence in X then a morphism

$$\begin{align*}F\colon (X,M_X)\to (Y,T_Y) \end{align*}$$

now can be seen as a generalisation of a normal lax functor. The identities are still preserved and the thin triangles of $M_X$ represent 2-morphisms in X, which tells us that F is “strictly” functorial on mapping categories. However, in this situation composability only holds up to chosen coherent 2-morphisms. As a consequence, the Grothendieck construction in the case of non-fibrant base provides a potent tool to understand normal lax $(\infty ,2)$ -functors in terms of their associated fibrations.

1.3. Variances

The zoo of $(\infty ,1)$ -categorical Grothendieck constructions is complicated by the fact that such Grothendieck constructions come in two variances. One can either consider the aforementioned Cartesian fibrations of $(\infty ,1)$ -categories over , or consider coCartesian fibrations over . The former correspond to $(\infty ,1)$ -functors

whereas the latter correspond to $(\infty ,1)$ -functors

Additionally, if one treats the case of functors

valued in $\infty $ -groupoids (spaces), one obtains more restrictive variants of Cartesian/coCartesian fibrations, called right fibrations and left fibrations, respectively, in [23, Ch. 2].

Each variance can be obtained from the other by appropriate dualisation proceedures, and so, in practice it is only necessary to prove one correspondence to obtain the other. In the world of $(\infty ,2)$ -categories, where there are four possible variances, a similar principle applies, although the dualisation procedures can become more complicated. As a result, we have focused on a single variance in our exploration of the $(\infty ,2)$ -categorical Grothendieck construction. Later in the introduction we will give a more complete account of known Grothendieck constructions, as well as a table of relations between them.

1.4. The $(\infty ,2)$ -categorical Grothendieck construction

The present paper provides a complete $(\infty ,2)$ -categorical Grothendieck construction. Loosely speaking, for every scaled simplicial set S, we provide an equivalence of $(\infty ,2)$ -bicategories (or simply $\infty $ -bicategories as in [25])

between 2-Cartesian fibrations² over S, and $(\infty ,2)$ -functors with values in $\infty $ -bicategories.

To understand this construction on an intuitive level, it is helpful to first consider the strict 2-categorical variant of the construction, developed by Buckley in [8]. In this setting, we consider strict 2-functors $p:\mathbb {C}\to \mathbb {D}$ . A 1-morphism $f:c\to \overline {c}$ in $\mathbb {C}$ is called Cartesian if, for every $a\in \mathbb {C}$ there is a homotopy pullback square of categories

A 2-morphism $\alpha :f\Rightarrow g$ in $\mathbb {C}(x,y)$ is called coCartesian if it is a coCartesian 1-morphism for the map

The functor P is then called a 2-Cartesian fibration if it admits Cartesian lifts of all 1-morphisms, and coCartesian lifts of all 2-morphisms.

In our previous paper, [5], we provided a model structure which we claimed modelled the appropriate $(\infty ,2)$ -categorical variant of the above definitions. To keep track of the data of (1) invertible 2-morphisms, (2) Cartesian 1-morphisms and (3) coCartesian 2-morphisms in the simplicial setting, we considered a 3-part decoration on simplicial sets. Given a simplicial set $X\in {\operatorname {Set}}_\Delta $ , we define a marking and biscaling on X to consist of

• • As in [25], invertible 2-morphisms are encoded as a collection $T_X\subset X_2$ of 2-simplices, which is required to contain degenerate simplices. The 2-simplices in $T_X$ are called thin 2-simplices.

• • As in [23], Cartesian 1-morphisms are encoded as a collection $M_X\subset X_1$ of 1-simplices, which is required to contain the degenerate 1-simplices. The 1-simplices in $M_X$ are called marked 1-simplices.

• • The coCartesian 2-morphisms are encoded as a collection $C_X\subset X_2$ . Since every invertible 2-morphism should be coCartesian, we require that $T_X\subset C_X$ . We refer to the 2-simplices in $C_X$ as lean 2-simplices.

A tuple $(X,M_X,T_X\subset C_X)$ is referred to as a marked-biscaled simplicial set (or $\textbf {MB}$ simplicial set for short). We denote the category of $\textbf {MB}$ simplicial sets by ${\operatorname {Set}}_{\Delta }^{\mathbf {mb}}$ . Summarising the main results from our paper [5], we have

Theorem. Let $(S,T_S)$ be a scaled simplicial set.

1. 1. There is a left proper, combinatorial, simplicial model structure on $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/(S,\sharp ,T_S\subset \sharp )}$ , called the 2-Cartesian model structure.

2. 2. If $S=\Delta ^0$ is the terminal scaled simplicial set, the resulting model structure models $\infty $ -bicategories.

3. 3. If $(S,T_S)$ is the scaled nerve of a strict 2-category $\mathbb {D}$ , every 2-Cartesion fibration of strict 2-categories $P:\mathbb {C}\to \mathbb {D}$ gives rise to a fibrant object of $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/(S,\sharp ,T_S\subset \sharp )}$ .

In the second of these results, our decoration becomes highly redundant. In a fibrant object, the marked 1-morphisms correspond to equivalences, the thin 2-simplices correspond to invertible 2-morphisms, but the lean 2-simplices are identical to the thin 2-simplices. To simplify our later computations, we rectify this redundancy by also considering marked-scaled simplicial sets, i.e., triples $(X,M_X,T_X)$ consisting of a simplicial set X, a collection of marked 1-simplices $M_X$ , and a collection of thin 2-simplices $T_X$ . The category of marked-scaled simplicial sets is denoted by ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ . The first result of this paper formalises the fact that marked-scaled simplicial sets should also model $\infty $ -bicategories.

Theorem. There is a left proper, combinatorial, ${\operatorname {Set}}_\Delta ^+$ -enriched model structure on ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ . Moreover, it is Quillen equivalent to the 2-Cartesian model structure on ${\operatorname {Set}}_{\Delta }^{\mathbf {mb}}$ , and thus models $\infty $ -bicategories.

This can be found in the body of the paper as Theorem 2.46 and Proposition 2.59.

The main construction of this paper yields a functor for each scaled simplicial set S

called the bicategorical straightening over S. The functor itself is simply a more highly decorated version of previous straightening functors (e.g., that of [23]), and is discussed in detail at the beginning of section 3. We then show that $\mathbb {S}\!{\operatorname {t}}_S$ admits a right adjoint $\mathbb {U}\!{\operatorname {n}}_S$ which we call the (bicategorical) unstraightening over S. As already discussed, the category $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ carries a model structure which models 2-Cartesian fibrations. If we equip the category of ${\operatorname {Set}}_\Delta ^+$ -enriched functors $\mathfrak {C}[S]^{\operatorname {op}} \to {{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ with the projective model structure, we obtain an enriched model category which models the $(\infty ,1)$ -category of $(\infty ,2)$ -functors . The main technical result of this paper is that this adjunction is in fact a Quillen equivalence.

Theorem (Theorem 3.81).

Let S be a scaled simplicial set. Then the bicategorical straightening-unstraightening adjunction defines a Quillen equivalence

between the model structure on (outer) 2-Cartesian fibrations over S and the projective model structure on ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functors $\mathfrak {C}[S]^{\operatorname {op}} \to {{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ with values in marked-scaled simplicial sets.

Observe that both model categories are in fact ${\operatorname {Set}}^+_\Delta $ -enriched categories. After performing elementary explicit verifications we prove that the functor $\mathbb {U}\!{\operatorname {n}}_S$ is compatible with the (co)tensoring yielding an upgrade of the previous theorem to an intrinsically bicategorical result.

Theorem (Corollary 3.86).

The bicategorical straightening is a left Quillen equivalence for any scaled simplicial set S. Moreover, the functor $\mathbb {U}\!{\operatorname {n}}_S$ provides an equivalence of $(\infty ,2)$ -categories

The majority of this paper is thus devoted to this proof. We will recapitulate the major ideas of the proof, as well as the structure of the paper, in the penultimate section of the introduction.

1.5. A relative 2-nerve

Although it is desirable to have a bicategorical Grothendieck construction that works in the most general context possible, many practical applications make use of those $\infty $ -bicategories which arise as scaled nerves of strict 2-categories. We provide a version of the Grothendieck construction better suited to this particular situation in the final section of this paper. In this context, we define an explicit version of the unstraightening functor over ${\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})$ , which we call the relative 2-nerve.

Theorem (Corollary 4.20).

Let $\mathbb {C}$ be a strict 2-category. Then there is a Quillen equivalence

and an equivalence of left-derived functors .

As in the $(\infty ,1)$ -categorical setting (see Section 3.2.5 in [23]) the benefits of a relative nerve construction are twofold: on the one hand, the relative 2-nerve is particularly computationally tractable and well-suited to explicit examples; on the other, the relative 2-nerve allows us to compare our $\infty $ -bicategorical Grothendieck construction to preexisting strict Grothendieck constructions. We apply our relative nerve construction to obtain a comparison with the Grothendieck construction appearing in [8]. The strict 2-categorical Grothendieck construction of [8] takes the form of an equivalence

for a 2-category $\mathbb {C}$ . The final result of the paper shows that relative 2-nerve coincides with $\mathbb {E}\!\operatorname {l}$ for every strict 2-functor with values in $2$ -categories.

Theorem (Theorem 4.21).

Let

be a 2-functor, and let $\tilde {F}$ denote the composite

Then there is an equivalence

of 2-Cartesian fibrations over ${\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})$ .

1.6. The zoo of Grothendieck constructions

To aid the reader in connecting our work here to other results in the literature, we here provide a brief overview of existing Grothendieck constructions, as well as known connections among them. Where practicable, we will choose the version of the construction with the correct variance to agree with our construction.

• • The classical Grothendieck construction of [18] takes the form of a equivalence

  
  for a 1-category C

• • The classical Grothendieck construction is often restricted to categories fibred in groupoids, in which case it takes the form of an equivalence

  
  for a 1-category C.

• • The strict 2-categorical Grothendieck construction of [8] takes the form of an equivalence

  
  for a 2-category $\mathbb {C}$ .

• • The three Grothendieck-Lurie constructions:

  1. – For $(\infty ,1)$ -categories fibred in $\infty $ -groupoids, the construction of [23, Ch 2] takes the form of a left Quillen equivalence

     
     for S a simplicial set. Here $({\operatorname {Set}}_\Delta )_{/S}$ is equipped with the model structure for right fibrations, and $({\operatorname {Set}}_\Delta )^{\mathfrak {C}[S]^{\operatorname {op}}}$ is equipped with the projective model structure obtained from the Kan-Quillen model structure.

  2. – For $(\infty ,1)$ -categories fibred in $(\infty ,1)$ -categories, the construction of [23, Ch. 3] takes the form of a left Quillen equivalence

     
     where S is a simplicial set. Here $({{\operatorname {Set}}_{\Delta }^+})_{/S}$ carries the Cartesian model structure and $({{\operatorname {Set}}_{\Delta }^+})^{\mathfrak {C}[S]^{\operatorname {op}}} $ carries the projective model structure on ${\operatorname {Set}}_\Delta $ -enriched functors.

  3. – For $\infty $ -bicategories fibred in $(\infty ,1)$ -categories, the construction of [25] takes the form of a left Quillen equivalence

     
     where S is a scaled simplicial set. Here $({{\operatorname {Set}}_{\Delta }^+})_{/S}$ carries the $\mathfrak {P}$ -anodyne model structure of [25, Section 3.2] and $({\operatorname {Set}}_\Delta ^+)^{{\mathfrak {C}}^{{\operatorname {sc}}}[S]^{({\operatorname {op}},{\operatorname {op}})}}$ carries the projective model structure on ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functors.

• • A general lax Grothendieck construction is given by Loubaton in [21, Theorem 6.1.4.2.], for $(\infty ,\omega )$ -categories. Corollary 6.1.4.3 of op. cit. provides the non-lax version. This later takes the form of an equivalence of $(\infty ,\omega )$ -categories

  $$\begin{align*}\underline{\operatorname{Hom}}(A,\underline{\omega}) \simeq \underline{\operatorname{LCart}} (A^\sharp) \end{align*}$$
  for any $(\infty ,\omega )$ -category A, where the left-hand side is the $(\infty ,\omega )$ -category of functors, and the right-hand side is the $(\infty ,\omega )$ -category of left Cartesian fibrations over A. We are not aware of a comparison to other models, though Loubaton also proves a lax colimit formula for this Grothendieck construction which might allow for a general comparison result.

• • The comprehension construction, defined by Riehl and Verity in [31] works in an $\infty $ -cosmos . In the $\infty $ -cosmos of quasi-categories, it provides a functor

  
  from functors into small quasi-categories to coCartesians fibrations over B. However, the authors defer the proof that this is an equivalence to a latter work.

• • A general $(\infty ,n)$ -categorical Grothendieck construction formulated in the language of n-fold complete Segal spaces is provided by Nuiten in [28], working mostly model-independently. This provides equivalences of $(\infty ,n+1)$ -categories

  
  natural in an $(\infty ,n+1)$ -category , where the right-hand side denotes the $(\infty,n+1)$ -category of what Nuiten terms n-coCartesian fibrations. Nuiten additionally proves a unicity theorem, showing that the n-category of such natural equivalences is contractible. The comparison to other models is thus reduced to showing that the definitions of the left- and right-hand side $n+2$ -functors coincide with Nuiten’s.

We summarise the known relations between these constructions in the following diagram. An arrow in the diagram represents a special case, e.g. $\operatorname {El}\to \operatorname {St}_S^+$ means that $\operatorname {El}$ is known to be equivalent to a special case of $\operatorname {St}_S^+$ . A dashed arrow will represent a relation which we conjecture to hold.

In particular:

1. 1. These relations are proven in [23]. In particular, the comparison of the $(\infty ,1)$ -categorical and strict cases passes through the relative nerve of section 3.2.5.

2. 2. In [25, Remark 4.5.10] the author claims that due to the formal differences in the construction of both straightening functors, no direct comparison seems possible. Instead, the author proves a comparison on fibrant object without showing naturality in [25, Prop. 4.5.10] using model-independent arguments.

   However, we attribute the difficulty of a comparison to the fact that both straightening functors have different variances. It follows from our construction (see point 4 on this list) of $\mathbb {S}\!{\operatorname {t}}_S$ that $\operatorname {St}^+_S$ morally has an outer Cartesian variance. Since $\operatorname {St}^+_S$ models the Grothendieck construction for $\infty $ -categories this construction is blind to the variance in 2-morphisms and it is seen as having simply a Cartesian variance. The construction of [25, Prop. 4.5.9] passes through two 1-morphism dualisations to obtain a Cartesian variant of $\operatorname {St}^{(2,1)}_S$ . We thus believe a more complete comparison result with the $\operatorname {St}_S^{(2,1)}$ after taking the pertinent 2-morphism dual, as well.

3. 3. We show this relation in Theorem 4.21, making use of a relative 2-nerve which we construct for that purpose in the final section.

4. 4. This relation is nearly immediate from the definitions. We give a proof in Proposition 3.13.

5. 5. We believe that this relation will hold, however, the constructions $\operatorname {St}_S^{(2,1)}$ and $\mathbb {S}\!{\operatorname {t}}_S$ are related by a 2-morphism dualisation. Given the difficulties inherent in realising 2-morphism duals in scaled simplicial sets, we defer any attempt to prove this statement for the time being.

   

1.7. Structure of the paper

In Section 2, we provide some background on the model structures and constructions that the paper will use. In particular, we describe model structures on scaled, marked-scaled and marked-biscaled simplicial sets and show that they define Quillen equivalent models for $\infty $ -bicategories. We also recall the model structure for 2-Cartesian fibrations from our previous paper [5].

Section 3 contains the main results and technical arguments of the paper, in particular the $(\infty ,2)$ -categorical Grothendieck construction. We first construct the adjunction which will define the Grothendieck construction, and show that it is a Quillen adjunction. To show that our straightening functor $\mathbb {S}\!{\operatorname {t}}_S$ is a Quillen equivalence, we perform the kind of dimensional induction used in [23, Section 3.2]:

• • We prove that $\mathbb {S}\!{\operatorname {t}}_{\Delta ^0}$ is a left Quillen equivalence in Section 3.4. The proof of this fact is quite direct, and proceeds by constructing a natural equivalence from $\mathbb {S}\!{\operatorname {t}}_{\Delta ^0}$ to a more canonical left Quillen equivalence $L\colon {\operatorname {Set}}_{\Delta }^{\mathbf {mb}}\to {{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ (which is defined in subsection 2.3).

• • We then prove that $\mathbb {S}\!{\operatorname {t}}_{\Delta ^n_\flat }$ is a left Quillen equivalence in subsection 3.5. This is the most technically demanding step in the proof. We first show that, for any 2-Cartesian fibration $p:X\to \Delta ^n_\flat $ , there is a homotopy pushout diagram

  

  in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/\Delta ^n_\flat }$ , where we are denoting by $X_n$ the fibre over $n\in \Delta ^n_\flat $ . This statement appears as Corollary 3.73, the proof of which occupies most of Section 3.5.

  Once the homotopy pushout is established, we can show that, for a 2-Cartesian fibration $X\to \Delta _\flat $ with i-fibre $X_i$ , the canonical map $\mathbb {S}\!{\operatorname {t}}_{\Delta ^0}(X_i)\to \mathbb {S}\!{\operatorname {t}}_{\Delta ^n_\flat }(X)(i)$ is an equivalence. We then use the fibrewise nature of equivalences in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/\Delta ^n_\flat }$ together with the pointwise nature of equivalences in $({{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }})^{{\mathfrak {C}}^{{\operatorname {sc}}}[\Delta ^n_\flat ]^{\operatorname {op}}}$ to complete the proof.

• • The general case follows from the special cases over simplices by an inductive argument nearly identical to that of [25, Prop. 3.8.4].

The final section of the paper, Section 4, is devoted to a relative nerve construction, and a comparison with the strict 2-categorical constructions of [8].

2. Preliminaries

We here collect background information and preliminary results which will be necessary for our arguments in the rest of the paper. While we will review some 2-category theory, it is impracticable to recapitulate all of the needed background on $(\infty ,2)$ -categories, so we will limit ourselves to a brief discussion of scaled simplicial sets, and direct the reader to [25] for more comprehensive background.

2.1. 2-categories

We will often use strict 2-categories as a tool to explore $\infty $ -bicategories. In this section, we briefly collect some notations and constructions we will use in the sequel.

Notation. By a 2-category, we will always mean a strict 2-category. By a 2-functor, we will mean a strict 2-functor unless specified otherwise. We will denote strict 2-categories by blackboard bold letters, e.g. $\mathbb {D}$ .

A 2-category $\mathbb {D}$ has three duals, determined by reversing the direction of 1-morphisms, 2-morphisms, or both, respectively. In this work, we will primarily make use of the 1-morphism dual. To better accord with the notation used in (scaled) simplicial sets, we will denote the 1-morphism dual simply by $\mathbb {D}^{\operatorname {op}}$ . Where needed, we denote the 2-morphism dual by $\mathbb {D}^{(-,{\operatorname {op}})}$ , and the dual which reverses both 1- and 2-morphisms by $\mathbb {D}^{({\operatorname {op}},{\operatorname {op}})}$ . We will denote the 1-category of 2-categories and strict 2-functors by $2\!\operatorname {Cat}$ .

Definition 2.1. Let I be a linearly ordered finite set. We define a $2$ -category ${\mathbb {O}}^{I}$ as follows

• • the objects of ${\mathbb {O}}^I$ are the elements of I,

• • the category ${\mathbb {O}}^{I}(i,j)$ of morphisms between objects $i,j \in I$ is defined as the poset of finite sets $S \subseteq I$ such that $\min (S)=i$ and $\max (S)=j$ ordered by inclusion,

• • the composition functors are given, for $i,j,l\in I$ , by

  $$\begin{align*}{\mathbb{O}}^{I}(i,j) \times {\mathbb{O}}^{I}(j,l) \to {\mathbb{O}}^{I}(i,l), \quad (S,T) \mapsto S \cup T. \end{align*}$$

When $I=[n]$ , we denote ${\mathbb {O}}^I$ by ${\mathbb {O}}^n$ . Note that the ${\mathbb {O}}^n$ form a cosimplicial object in $2\!\operatorname {Cat}$ , which we denote by ${\mathbb {O}}^\bullet $ .

Definition 2.2. Let $f:\mathbb {C} \to \mathbb {D}$ be a functor of 2-categories. Given an object $d \in \mathbb {D}$ we define the lax slice 2-category as follows:

• • Objects are morphisms $u:d \to f(c)$ in $\mathbb {D}$ with source d such that $c \in \mathbb {C}$ .

• • A 1-morphism from $u: d \to f(c)$ to $v:d \to f(c')$ is given by a 1-morphism $\alpha : c \to c'$ in $\mathbb {C}$ and a 2-morphism .

• • A 2-morphism in is given by a 2-morphism $\varepsilon : \alpha \Rightarrow \beta $ such that the diagram below commutes

  

  If the functor f is the identity on the 2-category $\mathbb {D}$ we will use the notation .

Example 2.3. Let I be a linearly ordered finite set and denote its minimum by i. Unraveling Definition 2.2 we see that is the 2-category given by:

• • Objects are subsets $S \subseteq I$ such that $\min (S)=i$ .

• • A morphism $S \to T$ is given by a subset $U \subseteq I$ such that $\max (S)=\min (U)$ and $\max (U)=\max (T)$ and such that $S \cup U \subseteq T$ .

• • We have a 2-morphism precisely if $U \subseteq V$ .

Remark 2.4. We observe that the non-empty mapping categories in are all contractible since each has an initial object.

To represent 2-categories as simplicial sets, we need a nerve operation. We denote the category of simplicial sets by ${\operatorname {Set}}_\Delta $ .

Definition 2.5. Given $\mathbb {D}\in 2\!\operatorname {Cat}$ , we define a simplicial set ${\operatorname {N}}_2(\mathbb {D})\in {\operatorname {Set}}_\Delta $ , the Duskin nerve of $\mathbb {D}$ , by

$$\begin{align*}{\operatorname{N}}_2(\mathbb{D})_n:= 2\!\operatorname{Cat}({\mathbb{O}}^n,\mathbb{D}). \end{align*}$$

2.2. Higher categories and decorated simplicial sets

Throughout this paper, we will make extensive use of models for higher categories in terms of decorated simplicial sets. For models for $(\infty ,1)$ -categories, we will direct the reader to [23, §2.2.5, §3.1], though we briefly discuss notation here.

Definition 2.6. We denote the category of simplicial sets by ${\operatorname {Set}}_\Delta $ . A marked simplicial set is defined to be a pair $(X,M_X)$ consisting of a simplicial set $X\in {\operatorname {Set}}_\Delta $ , and a collection $M_X\subseteq X_1$ of 1-simplices in X which contains the degenerate 1-simplices. We will denote the category of marked simplicial sets by ${{\operatorname {Set}}_{\Delta }^+}$ .

We will typically view ${\operatorname {Set}}_\Delta $ as equipped with either the Kan-Quillen model structure (see, e.g., [16, Ch. 1]) or the Joyal model structure (see, e.g. [23, §2.2.5]). We will view ${{\operatorname {Set}}_{\Delta }^+}$ as equipped with the Cartesian model structure of [23, §3.1].

The first model structure we will use to study $\infty $ -bicategories is a model structure on enriched categories.

Notation. We denote by $\operatorname {Cat}_\Delta ^+$ the category of ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories.

Proposition 2.7. There is a left-proper, combinatorial model structure on $\operatorname {Cat}_\Delta ^+$ such that

1. W The weak equivalences are those enriched functors which are essentially surjective on homotopy categories and induce equivalences on all mapping spaces.

2. C The cofibrations are the smallest weakly saturated class containing $\varnothing \to [0]_{{{\operatorname {Set}}_{\Delta }^+}}$ , and each inclusion $[1]_A\to [1]_{B}$ where $A\to B$ is a generating cofibration for ${{\operatorname {Set}}_{\Delta }^+}$ .

Proof. This is a special case of [23, A.3.2.4].

Remark 2.8. Notice that, given a strict 2-category $\mathbb {D}$ , we can take the nerves of the mapping categories to obtain a ${\operatorname {Set}}_\Delta $ -enriched category. If we define a marking on each mapping category by declaring precisely the isomorphisms to be marked, we obtain a canonical element of $\operatorname {Cat}_\Delta ^+$ associated to $\mathbb {D}$ . We will uniformly abuse notation by denoting this ${{\operatorname {Set}}_{\Delta }^+}$ -enriched category by $\mathbb {D}$ as well.

The basic idea for the main models for $\infty $ -bicategories used in this paper is that simplicial sets may be used to model $\infty $ -bicategories, provided we keep track of which 2-simplices are considered to represent invertible 2-morphisms.

Definition 2.9. A scaled simplicial set consists of a pair $(X,T_X)$ , where $X\in {\operatorname {Set}}_\Delta $ is a simplicial set, and $T_X\subseteq X_2$ is a collection of 2-simplices – called the thin 2-simplices – which contains all degenerate 2-simplices. We denote by ${{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ the category of scaled simplicial sets.

Notation. We will sometimes make use of subscripts to denote scalings. In particular, $(X,X_2)$ will be will be denoted by $X_\sharp $ , and $(X,\operatorname {deg}(X_2))$ will be denoted by $X_\flat $ . Similarly, we will sometimes use superscripts to denote markings on a simplicial set: $(X,X_1)=X^\sharp $ , and $(X,\operatorname {deg}(X_1))=X^\flat $ .

Definition 2.10. The set of generating scaled anodyne maps S is the set of maps of scaled simplicial sets consisting of:

1. (i) the inner horns inclusions

   $$\begin{align*}\bigl(\Lambda^n_i,\{\Delta^{\{i-1,i,i+1\}}\}\bigr)\rightarrow \bigl(\Delta^n,\{\Delta^{\{i-1,i,i+1\}}\}\bigr), \quad n \geqslant 2, \quad 0 < i < n; \end{align*}$$

2. (ii) the map

   $$\begin{align*}(\Delta^4,T)\rightarrow (\Delta^4,T\cup \{\Delta^{\{0,3,4\}}, \ \Delta^{\{0,1,4\}}\}), \end{align*}$$

   where we define

   $$\begin{align*}T\overset{\text{def}}{=}\{\Delta^{\{0,2,4\}}, \ \Delta^{\{ 1,2,3\}}, \ \Delta^{\{0,1,3\}}, \ \Delta^{\{1,3,4\}}, \ \Delta^{\{0,1,2\}}\}; \end{align*}$$

3. (iii) the set of maps

   $$\begin{align*}\Bigl(\Lambda^n_0\coprod_{\Delta^{\{0,1\}}}\Delta^0,\{\Delta^{\{0,1,n\}}\}\Bigr)\rightarrow \Bigl(\Delta^n\coprod_{\Delta^{\{0,1\}}}\Delta^0,\{\Delta^{\{0,1,n\}}\}\Bigr), \quad n\geqslant 3. \end{align*}$$

A general map of scaled simplicial set is said to be scaled anodyne if it belongs to the weakly saturated closure of S.

Definition 2.11. We say that a map of scaled simplicial sets $p:X \to S$ is a weak S-fibration if it has the right lifting property with respect to the class of scaled anodyne maps. We call $X\in {{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ a $\infty $ -bicategory if the unique map $X\to \Delta ^0_\sharp $ is a weak S-fibration.

Definition 2.12. The composite

gives us a cosimplicial object in $\operatorname {Cat}_\Delta ^+$ . We can moreover send the thin 2-simplex $\Delta ^2_\sharp $ to $\mathfrak {C}[\Delta ^2]$ equipped with sharp-marked mapping spaces. The usual machinery of nerve and realisation then gives us adjoint functors

which we will call the scaled nerve and scaled rigidification.

Remark 2.13. Given a 2-category $\mathbb {D}$ , we can define a scaling on $N_2(\mathbb {D})$ by declaring a triangle to be thin if and only if the corresponding 2-morphism is invertible. Viewing $\mathbb {D}$ as an ${{\operatorname {Set}}_{\Delta }^+}$ -enriched category as in Remark 2.8, we see that this scaled simplicial set coincides with ${\operatorname {N}}^{\operatorname {sc}}(\mathbb {D})$ . We thus are justified in speaking of the scaled nerve of a 2-category.

Theorem 2.14. There is a left proper, combinatorial model structure on ${{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ with

1. W The weak equivalences are the morphisms $f:A\to B$ such that ${\mathfrak {C}}^{{\operatorname {sc}}}[f]:{\mathfrak {C}}^{{\operatorname {sc}}}[A]\to {\mathfrak {C}}^{{\operatorname {sc}}}[B]$ is an equivalence in $\operatorname {Cat}_\Delta ^+$ .

2. C The cofibrations are the monomorphisms.

Moreover, the fibrant objects in this model structure are the $\infty $ -bicategories, and the adjunction

is a Quillen equivalence.

Proof. This is [25, Thm A.3.2.4]. The characterisation of fibrant objects is[11, Thm 5.1].

Remark 2.15 (Key notational convention).

When no confusion is likely to arise, we denote decorated simplicial sets by simple roman majescules: X, Y, Z, etc. For fibrant objects, representing (higher) categories of various kinds, we fix the following conventions:

• • Strict 1-categories will be denoted by undecorated roman majescules, B, C, D, etc.

• • $(\infty ,1)$ -categories as presented by Joyal fibrant simplicial sets or fibrant marked simplicial sets will be denoted by calligraphic majescules: , , , etc.

• • Strict 2-categories will be denoted by blackboard-bold majescules, $\mathbb {B}$ , $\mathbb {C}$ , $\mathbb {D}$ , etc.

• • $(\infty ,2)$ -categories, presented as fibrant scaled simplicial sets, will be denoted by thickened blackboard-bold majescules: , , , etc.

2.3. Marked biscaled simplicial sets and 2-Cartesian fibrations.

In this section we collect useful definitions and results introduced in [5] that will play a relevant role in this paper.

Definition 2.16. A marked biscaled simplicial set (mb simplicial set) is given by the following data

• • A simplicial set X.

• • A collection of edges $E_X \in X_1$ containing all degenerate edges.

• • A collection of triangles $T_X \in X_2$ containing all degenerate triangles. We will refer to the elements of this collection as thin triangles.

• • A collection of triangles $C_X \in X_2$ such that $T_X \subseteq C_X$ . We will refer to the elements of this collection as lean triangles.

We will denote such objects as triples $(X,E_X, T_X \subseteq C_X)$ . A map $(X,E_X, T_X \subseteq C_X) \to (Y,E_Y,T_Y \subseteq C_Y)$ is given by a map of simplicial sets $f:X \to Y$ compatible with the collections of edges and triangles above. We denote by ${\operatorname {Set}}_{\Delta }^{\mathbf {mb}}$ the category of mb simplicial sets.

Notation. Let $(X,E_X, T_X \subseteq C_X)$ be a mb simplicial set. Suppose that the collection $E_X$ consist only of degenerate edges. Then we fix the notation $(X,E_X, T_X \subseteq C_X)=(X,\flat ,T_X \subseteq E_X)$ and do similarly for the collection $T_X$ . If $C_X$ consists only of degenerate triangles we fix the notation $(X,E_X, T_X \subseteq C_X)=(X,E_X, \flat )$ . In an analogous fashion we wil use the symbol “ $\sharp $ ” to denote a collection containing all edges (resp. all triangles). Finally suppose that $T_X=C_X$ then we will employ the notation $(X,E_X,T_X)$ .

Remark 2.17. We will often abuse notation when defining the collections $E_X$ (resp. $T_X$ , resp. $C_X$ ) and just specify its non-degenerate edges (resp. triangles).

Definition 2.18. The set of generating mb anodyne maps MB is the set of maps of mb simplicial sets consisting of:

1. (A1) The inner horn inclusions

   $$\begin{align*}\bigl(\Lambda^n_i,\flat,\{\Delta^{\{i-1,i,i+1\}}\}\bigr)\rightarrow \bigl(\Delta^n,\flat,\{\Delta^{\{i-1,i,i+1\}}\}\bigr), \quad n \geqslant 2, \quad 0 < i < n; \end{align*}$$

2. (A2) The map

   $$\begin{align*}(\Delta^4,\flat,T)\rightarrow (\Delta^4,\flat,T\cup \{\Delta^{\{0,3,4\}}, \ \Delta^{\{0,1,4\}}\}), \end{align*}$$

   where we define

   $$\begin{align*}T\overset{\text{def}}{=}\{\Delta^{\{0,2,4\}}, \ \Delta^{\{ 1,2,3\}}, \ \Delta^{\{0,1,3\}}, \ \Delta^{\{1,3,4\}}, \ \Delta^{\{0,1,2\}}\}; \end{align*}$$

3. (A3) The set of maps

   $$\begin{align*}\Bigl(\Lambda^n_0\coprod_{\Delta^{\{0,1\}}}\Delta^0,\flat,\flat \subset\{\Delta^{\{0,1,n\}}\}\Bigr)\rightarrow \Bigl(\Delta^n\coprod_{\Delta^{\{0,1\}}}\Delta^0,\flat,\flat \subset\{\Delta^{\{0,1,n\}}\}\Bigr), \quad n\geqslant 2. \end{align*}$$

   These maps force left-degenerate lean-scaled triangles to represent coCartesian edges of the mapping category.

4. (A4) The set of maps

   $$\begin{align*}\Bigl(\Lambda^n_n,\{\Delta^{\{n-1,n\}}\},\flat \subset \{ \Delta^{\{0,n-1,n\}} \}\Bigr) \to \Bigl(\Delta^n,\{\Delta^{\{n-1,n\}}\},\flat \subset \{ \Delta^{\{0,n-1,n\}} \}\Bigr), \quad n \geqslant 2. \end{align*}$$

   This forces the marked morphisms to be p-Cartesian with respect to the given thin and lean triangles.

5. (A5) The inclusion of the terminal vertex

   $$\begin{align*}\Bigl(\Delta^{0},\sharp,\sharp \Bigr) \rightarrow \Bigl(\Delta^1,\sharp,\sharp \Bigr). \end{align*}$$

   This requires p-Cartesian lifts of morphisms in the base to exist.

6. (S1) The map

   $$\begin{align*}\Bigl(\Delta^2,\{\Delta^{\{0,1\}}, \Delta^{\{1,2\}}\},\sharp \Bigr) \rightarrow \Bigl(\Delta^2,\sharp,\sharp \Bigr), \end{align*}$$

   requiring that p-Cartesian morphisms compose across thin triangles.

7. (S2) The map

   $$\begin{align*}\Bigl(\Delta^2,\flat,\flat \subset \sharp \Bigr) \rightarrow \Bigl( \Delta^2,\flat,\sharp\Bigr), \end{align*}$$

   which requires that lean triangles over thin triangles are, themselves, thin.

8. (S3) The map

   $$\begin{align*}\Bigl(\Delta^3,\flat,\{\Delta^{\{i-1,i,i+1\}}\}\subset U_i\Bigr) \rightarrow \Bigl(\Delta^3,\flat, \{\Delta^{\{i-1,i,i+1\}}\}\subset \sharp \Bigr), \quad 0<i<3 \end{align*}$$

   where $U_i$ is the collection of all triangles except the i-th face. This and the next two generators serve to establish composability and limited 2-out-of-3 properties for lean triangles.

9. (S4) The map

   $$\begin{align*}\Bigl(\Delta^3 \coprod_{\Delta^{\{0,1\}}}\Delta^0,\flat,\flat \subset U_0\Bigr) \rightarrow \Bigl(\Delta^3 \coprod_{\Delta^{\{0,1\}}}\Delta^0,\flat, \flat \subset \sharp \Bigr) \end{align*}$$

   where $U_0$ is the collection of all triangles except the $0$ -th face.

10. (S5) The map

    $$\begin{align*}\Bigl(\Delta^3,\{\Delta^{\{2,3\}}\},\flat \subset U_3\Bigr) \rightarrow \Bigl(\Delta^3,\{\Delta^{\{2,3\}}\}, \flat \subset \sharp \Bigr) \end{align*}$$

    where $U_3$ is the collections of all triangles except the $3$ -rd face.

11. (E) For every Kan complex K, the map

    $$\begin{align*}\Bigl( K,\flat,\sharp \Bigr) \rightarrow \Bigl(K,\sharp, \sharp\Bigr). \end{align*}$$

    Which requires that every equivalence is a marked morphism.

A map of mb simplicial sets is said to be MB-anodyne if it belongs to the weakly saturated closure of MB.

Definition 2.19. Let $f:(X,E_X,T_X \subseteq C_X) \to (Y,E_Y,T_Y \subseteq C_Y)$ be a map of mb simplicial sets. We say that f is a MB-fibration if it has the right lifting property against the class of MB-anodyne morphisms.

Definition 2.20. Given two mb simplicial sets $(K,E_K,T_K \subseteq C_K), (X,E_X,T_X \subseteq C_X)$ we define another mb simplicial set denoted by ${\operatorname {Fun}}^{\mathbf {mb}}(K,X)$ and characterised by the following universal property

$$\begin{align*}{\operatorname{Hom}}_{{\operatorname{Set}}_{\Delta}^{\mathbf{mb}}}\Bigl(A,{\operatorname{Fun}}^{\mathbf{mb}}(K,X) \Bigr)\simeq {\operatorname{Hom}}_{{\operatorname{Set}}_{\Delta}^{\mathbf{mb}}}\Bigl(A \times K,X \Bigr). \end{align*}$$

Proposition 2.21. Let $f:(X,E_X,T_X \subseteq C_X) \to (Y,E_Y,T_Y \subseteq C_Y)$ be a MB-fibration. Then for every $K \in {\operatorname {Set}}_{\Delta }^{\mathbf {mb}}$ the induced morphism ${\operatorname {Fun}}^{\mathbf {mb}}(K,X) \to {\operatorname {Fun}}^{\mathbf {mb}}(K,Y)$ is a MB-fibration.

Proof. This is [5, Cor. 3.17].

Definition 2.22. Let $g: K\to Y$ and $f:X\to Y$ be morphisms of mb simplicial sets. We define a mb simplicial set $\operatorname {Map}_Y(K,X)$ by means of the pullback square

If $f:X \to Y$ is a MB-fibration then it follows from the previous proposition that $\operatorname {Map}_Y(K,X)$ is an $\infty $ -bicategory.

Let $S \in {\operatorname {Set}}^{\mathbf {sc}}_{\Delta }$ be a scaled simplicial set. For the rest of the section we will denote $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ the category of mb simplicial sets over $(S,\sharp ,T_S \subset \sharp )$ .

Definition 2.23. We say that an object $\pi :X \to S$ in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ is an outer 2-Cartesian fibration if it is a $\textbf {MB}$ -fibration.

Remark 2.24. We will frequently abuse notation and refer to outer 2-Cartesian as 2-Cartesian fibrations.

Definition 2.25. Let $\pi :X \to S$ be a morphism of mb simplicial sets. Given an object $K\to S$ , we define ${\operatorname {Map}}^{\operatorname {th}}_{S}(K,X)$ to be the mb sub-simplicial set consisting only of the thin triangles. Note that if $\pi $ is a 2-Cartesian fibration this is precisely the underlying $\infty $ -category of $\operatorname {Map}_S(K,X)$ .

We similarly denote by ${\operatorname {Map}}^{\simeq }_S(K,X)$ the mb sub-simplicial set consisting of thin triangles and marked edges. As before, we note that if $\pi $ is a 2-Cartesian fibration, the simplicial set ${\operatorname {Map}}^{\simeq }_S(K,X)$ can be identified with the maximal Kan complex in $\operatorname {Map}_S(K,X)$ .

Definition 2.26. We define a functor $I\colon {\operatorname {Set}}^+_{\Delta } \to {\operatorname {Set}}_{\Delta }^{\mathbf {mb}}$ mapping a marked simplicial set $(K,E_K)$ to the mb simplicial set $(K,E_K,\sharp )$ . If K is maximally marked we adopt the notation $I^{+}(K^{\sharp })=K^{\sharp }_{\sharp }$

Remark 2.27. Note that we can endow the $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ with the structure of a ${\operatorname {Set}}_{\Delta }^{+}$ -enriched category by means of ${\operatorname {Map}}^{\operatorname {th}}_{S}(\mathord {-},\mathord {-})$ . In addition given $K \in {\operatorname {Set}}_{\Delta }^+$ and $\pi :X \to S$ we define $K \otimes X:= I(K) \times X$ equipped with a map to S given by first projecting to X and then composing with $\pi $ . This construction shows that $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ is tensored over ${\operatorname {Set}}^+_{\Delta }$ . One can easily show that $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ is also cotensored over ${\operatorname {Set}}^+_{\Delta }$ .

In a similar way one can use ${\operatorname {Map}}^{\simeq }_{S}(\mathord {-},\mathord {-})$ to endow $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ with the structure of a ${\operatorname {Set}}_{\Delta }$ -enriched category. In this case the tensor is given by $K \otimes X= I(K^{\sharp }) \times X$ .

Definition 2.28. Let be a morphism in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ . We say that h is a cofibration when it is a monomorphism of simplicial sets. We will call h a weak equivalence if for every 2-Cartesian fibration $\pi :X \to S$ the induced morphism

is a bicategorical equivalence.

Proposition 2.29. Suppose we are given a morphism of 2-Cartesian fibrations

Then the following are equivalent:

1. i) The map f is a weak equivalence.

2. ii) For every $s \in S$ the induced morphism on fibres $f_s \colon X_s \to Y_s$ is an equivalence of scaled simplicial sets.

Proof. See Proposition 3.38 in [5].

For the convenience of the reader, we here recall the main result of [5]:

Theorem 2.30 ([5] Theorem 3.38).

Let S be a scaled simplicial set. Then there exists a left proper combinatorial simplicial model structure on $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}} )_{/S}$ , which is characterised uniquely by the following properties:

1. C) A morphism $f:X \to Y$ in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}} )_{/S}$ is a cofibration if and only if f induces a monomorphism on the underlying simplicial sets.

2. F) An object $X \in ({\operatorname {Set}}_{\Delta }^{\mathbf {mb}} )_{/S}$ is fibrant if and only if X is a 2-Cartesian fibration.

2.3.1. MB-anodyne morphisms and dull subsets

Before proceeding, we here record two variants of the pivot point trick [6, Lem. 1.10] which will be of use later.

Definition 2.31. Let $\mathbb {P}(n)$ be the power set of $[n]$ . Given $\mathcal {A} \subset \mathbb {P}(n)$ and $X \in \mathbb {P}(n)$ we say that X is $\mathcal {A}$ -basal if X contains precisely one element from each $S\in \mathcal {A}$ and $X\subset \bigcup _{S\in \mathcal {A}}S$ . We denote the set of $\mathcal {A}$ -basal sets by $\operatorname {Bas}(\mathcal {A})$ .

Definition 2.32. Given a subset $\mathcal {A}\subset \mathbb {P}(n)$ such that $\varnothing \notin \mathcal {A}$ , and a marked-biscaled simplex $(\Delta ^n)^\dagger $ (where the dagger represents an arbitary marking and biscaling), we define a marked-biscaled simplicial subset

equipped with the marking and scaling inherited from $(\Delta ^n)^\dagger $ .

Definition 2.33. We call a subset $\mathcal {A}\subset \mathbb {P}(n)$ inner-dull with respect to a pivot point $0<i<n$ if the following conditions are satisfied

1. 1. $\mathcal {A}$ does not contain $\varnothing $ .

2. 2. We have that $i \notin S$ for every $S \in \mathcal {A}$ .

3. 3. For any $S,T \in \mathcal {A}$ such that $S\neq T$ , $S \cap T=\varnothing $ .

4. 4. For every $\mathcal {A}$ -basal set $X \in \mathbb {P}(n)$ there exist $u,v \in X$ such that $u<i < v$ .

Definition 2.34. Given an inner-dull subset $\mathcal {A} \subset \mathbb {P}(n)$ with pivot point i, we define to be the set of subsets $X \in \mathbb {P}(n)$ satisfying:

1. A1) X contains the pivot point $i \in X$ .

2. A2) The simplex $\sigma _X:\Delta ^X \to (\Delta ^n)^{\dagger }$ does not factor through $ (\mathcal {S}^{\mathcal {A}})^\dagger $ .

We define . Note that those elements of minimal cardinality are of the form for $X_0 \in \operatorname {Bas}(\mathcal {A})$ .

Definition 2.35. Let $\mathcal {A} \subset \mathbb {P}(n)$ be an inner-dull subset with pivot point i. Given an $\mathcal {A}$ -basal subset X we denote by $l^X < u^X$ the pair of consecutive elements in X such that $l^X <i < u^X$ .

Lemma 2.36 (The pivot trick).

Let $\mathcal {A} \subset \mathbb {P}(n)$ be an inner-dull subset with pivot point i and let $(\Delta ^n)^{\dagger }$ be a marked biscaled simplex. Suppose that the following conditions hold:

1. 1. Every marked edge (resp. thin triangle) which does not contain the pivot point i factors through $(\mathcal {S}^{\mathcal {A}})^\dagger $ .

2. 2. For every $X \in \operatorname {Bas}(\mathcal {A})$ and every $l^X\leqslant r <i < s \leqslant u^X$ the triangle is thin.

3. 3. Let be a lean simplex not containing the pivot point i. Then either $\sigma $ factors through $(\mathcal {S}^{\mathcal {A}})^\dagger $ or we have $a < i <c$ and the simplex is fully lean scaled.

Then the inclusion

$$\begin{align*}(\mathcal{S}^{\mathcal{A}})^\dagger \to (\Delta^n)^\dagger \end{align*}$$

is in the weakly saturated hull of morphisms of type (A1) and (S3).

Proof. Observe that since $\mathcal {A}$ is inner-dull it follows that every $\mathcal {A}$ -basal set has the same cardinality which we denote $\varepsilon $ . For every $\varepsilon \leqslant j \leqslant n$ we define

where $Y_{\varepsilon -1}=(\mathcal {S}^{\mathcal {A}})^\dagger $ and we view $\sigma _X$ as having the inherited decorations. This yields a filtration

$$\begin{align*}(\mathcal{S}^{\mathcal{A}})^\dagger \to Y_\varepsilon \to \cdots \to Y_{n-1} \to (\Lambda^n_i)^{\dagger} \to (\Delta^n)^\dagger \end{align*}$$

We will show that each step of this filtration can be obtained as an iterated pushout along morphisms of type 2.18 and 2.18. Let $X \in \mathcal {M}_{\mathcal {A}}^j$ for $\varepsilon \leqslant j \leqslant n-1$ and consider the pullback diagram

We claim that the top horizontal morphism is in the weakly saturated hull of morphisms of type 2.18 and 2.18. First we notice that the triangle is thin in $\Delta ^X$ by virtue of condition (2). Observe that if the dimension of $\Delta ^X$ is bigger than $3$ then all the possible decorations factor through $ \Lambda ^X_i$ . We will therefore assume that the dimension is at most 3, since otherwise the claim follows directly. Suppose that $\varepsilon =2$ so that we can have some $\Delta ^X$ of dimension $2$ . The edge of $\Delta ^X$ which does not contain i does not factor through $(\mathcal {S}^{\mathcal {A}})^\dagger $ , and so by condition (1) cannot be marked. If $\varepsilon =2$ and the dimension of $\Delta ^X$ is $3$ then it again follows by condition (1) that the face that misses the vertex i cannot be thin-scaled. If that face is not lean-scaled then the claim follows immediately. Otherwise our assumptions imply that $\Delta ^X$ is fully lean scaled, so that the map $ \Lambda ^X_i \to \Delta ^X$ is a composite of a morphism of type 2.18 and a morphism of type 2.18. The final case $\varepsilon =3$ is similar and left as an exercise.

We finish the proof by noting that it follows that $\sigma _X \cap \sigma _Y \in Y_{j-1}$ which implies that the order in which the add the simplices is irrelevant. We conclude that each step in the filtration belongs to the weakly saturated hull of morphisms of type 2.18 and 2.18.

We finish the discussion on dull subsets by giving a right-horn variant of the previous construction.

Definition 2.37. We call a subset $\mathcal {A}\subset \mathbb {P}(n)$ right-dull if the following conditions are satisfied

1. 1. $\mathcal {A}$ does not contain $\varnothing $ .

2. 2. For every $S\in \mathcal {A}$ , $n\notin S$ .

3. 3. For any $S,T\in \mathcal {A}$ such that $S\neq T$ , $S\cap T=\varnothing $ .

4. 4. For every $\mathcal {A}$ -basal subset X we have $u,v \in X$ such that $u<v<n$ .

In this case we call n the pivot point.

Lemma 2.38. Let $\mathcal {A}\subset \mathbb {P}(n)$ be a right-dull subset. Let $(\Delta ^n)^{\dagger }$ be a marked-biscaled simplex whose thin triangles are degenerate. Suppose that the following conditions hold

• • For every $\mathcal {A}$ -basal subset X and for every $s,r\in [n]$ such that $s\leqslant \min (X) < \max (X)\leqslant r<n$ , the triangle $\{s<r<n\}$ is lean, and the edge $r\to n$ is marked.

• • Let e be a marked edge in $(\Delta ^n)^{\dagger }$ not containing the vertex n. Then e factors through $(\mathcal {S}^{\mathcal {A}})^\dagger $ .

• • Let be a lean triangle in $(\Delta ^n)^{\dagger }$ not containing the vertex n. Then either $\sigma $ factors through $(\mathcal {S}^{\mathcal {A}})^\dagger $ or is fully lean-scaled and $c \to n$ is marked.

Then $(\mathcal {S}^{\mathcal {A}})^\dagger \to (\Delta ^n)^\dagger $ is in the saturated hull of morphisms of type (A4)

Proof. The argument is nearly identical to the proof of Lemma 2.36.

Lemma 2.39. Let $\mathcal {A}\subset \mathbb {P}(n)$ be a right-dull subset. Let $(\Delta ^n)^{\dagger }=(\Delta ^n,E_n,T_n \subset C_n)$ be a marked-biscaled simplex such that $(\Delta ^n)^\diamond :=(\Delta ^n,E_n,\flat \subset C_n)$ satisfies the hypothesis of Lemma 2.38. Suppose that we are given a morphism

$$\begin{align*}(\Delta^n,E_n,T_n \subset C_r) \to (X,\sharp,T_X \subset \sharp) \end{align*}$$

Then the morphism $(\mathcal {S}^{\mathcal {A}})^\dagger \to (\Delta ^n)^\dagger $ is an $\textbf {MB}$ -anodyne morphism over $(X,T_X)$ .

Proof. By Lemma 2.38 we obtain a pushout diagram

where the top horizontal morphism is $\textbf {MB}$ -anodyne. Note P only differs from $(\Delta ^n)^\dagger $ in its thin-scaling. Moreover every lean triangle in P whose image in $(\Delta ^n)^{\dagger }$ is thin gets mapped to a thin triangle in $(X,T_X)$ so it can be scaled using a morphism of type (S2).

2.4. Marked scaled simplicial sets

A special case of the model structure of Theorem 2.30 of particular interest occurs when $S=\Delta ^0$ is the terminal scaled simplicial set. Then, by [5, Thm 3.39], the resulting model structure on ${\operatorname {Set}}_{\Delta }^{\mathbf {mb}}$ is Quillen equivalent to the model structure for $\infty $ -bicategories on ${\operatorname {Set}}_\Delta ^{\mathbf {sc}}$ . In this case, the data of the two scalings becomes highly redundant – for any fibrant object the two scalings coincide, and heuristically they no longer encode different information.

We can avoid this redundancy by defining a further model structure which includes both markings and scalings, but avoids the redundancies created by a biscaling. The aim of this section is to define this model structure, and relate it to the MB model structure.

Definition 2.40. A marked-scaled simplicial set consists of

• • A simplicial set X.

• • A collection of edges $E_X\subseteq X_1$ containing all degenerate edges. We call the elements of $E_X$ marked edges.

• • A collection of triangles $T_X\subseteq X_2$ containing all degenerate triangles. We call the elements of $T_X$ thin triangles.

We denote by ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ the category of marked-scaled simplicial sets. We view this as a ${\operatorname {Set}}_\Delta ^+$ -enriched category by defining

$$\begin{align*}{\operatorname{Hom}}_{{\operatorname{Set}}_\Delta^+}(X,{{\operatorname{Set}}^{\mathbf{ms}}_{\Delta}}(Y,Z)):= {\operatorname{Hom}}_{{{\operatorname{Set}}^{\mathbf{ms}}_{\Delta}}}(X_\sharp\times Y,Z) \end{align*}$$

where $X_\sharp =(X,E_X,\sharp )$ .

Before continuing with the construction of the model structure, we briefly digress to explore the relations between ${\operatorname {Set}}_{\Delta }^{\mathbf {mb}}$ and ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ . The primary component of our comparison will be the adjunction:

where D is given on objects by

$$\begin{align*}D\colon (X,E_X,T_X) \mapsto (X,E_X,T_X\subseteq T_X) \end{align*}$$

and R is given on objects by

$$\begin{align*}R \colon(Y,E_Y,T_Y\subseteq C_Y) \mapsto (Y,E_Y,T_Y) \end{align*}$$

We will show that this adjunction becomes a Quillen equivalence once we have equipped ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ with the appropriate model structure.

This model structure itself is constructed exactly analogously to the model structure on ${\operatorname {Set}}_{\Delta }^{\mathbf {mb}}$ . We begin with a set of generating anodyne morphisms:

Definition 2.41. The set of generating $\mathbf {MS}$ -anodyne maps $\mathbf {MS}$ is the set of maps of marked-scaled simplicial sets consisting of:

1. (MS1) The inner horn inclusions

   $$\begin{align*}\bigl(\Lambda^n_i,\flat,\{\Delta^{\{i-1,i,i+1\}}\}\bigr)\rightarrow \bigl(\Delta^n,\flat,\{\Delta^{\{i-1,i,i+1\}}\}\bigr), \quad n \geqslant 2, \quad 0 < i < n; \end{align*}$$

2. (MS2) The map

   $$\begin{align*}(\Delta^4,\flat,T) \to (\Delta^4,\flat, T\cup \{\Delta^{\{0,3,4\}},\Delta^{\{0,1,4\}}\}) \end{align*}$$

   where T is defined as in Definition 2.18, (A2).

3. (MS3) The set of maps

   $$\begin{align*}\Bigl(\Lambda^n_0\coprod_{\Delta^{\{0,1\}}}\Delta^0,\flat,\{\Delta^{\{0,1,n\}}\}\Bigr)\rightarrow \Bigl(\Delta^n\coprod_{\Delta^{\{0,1\}}}\Delta^0,\flat,\{\Delta^{\{0,1,n\}}\}\Bigr), \quad n\geqslant 2. \end{align*}$$

4. (MS4) The set of maps

   $$\begin{align*}\Bigl(\Lambda^n_n,\{\Delta^{\{n-1,n\}}\}, \{ \Delta^{\{0,n-1,n\}} \}\Bigr) \to \Bigl(\Delta^n,\{\Delta^{\{n-1,n\}}\}, \{ \Delta^{\{0,n-1,n\}} \}\Bigr) , \quad n \geqslant 2. \end{align*}$$

5. (MS5) The inclusion of the terminal vertex

   $$\begin{align*}\left(\Delta^0,\sharp,\sharp\right)\to \left(\Delta^1,\sharp,\sharp\right) \end{align*}$$

6. (MS6) The map

   $$\begin{align*}\Bigl(\Delta^2,\{\Delta^{\{0,1\}}, \Delta^{\{1,2\}}\},\sharp \Bigr) \rightarrow \Bigl(\Delta^2,\sharp,\sharp \Bigr), \end{align*}$$

7. (MS7) The map

   $$\begin{align*}\Bigl(\Delta^3 \coprod_{\Delta^{\{0,1\}}}\Delta^0,\flat, U_0\Bigr) \rightarrow \Bigl(\Delta^3 \coprod_{\Delta^{\{0,1\}}}\Delta^0,\flat, \sharp \Bigr) \end{align*}$$

   where $U_0$ is the collection of all triangles except the $0$ -th face.

8. (MS8) The map

   $$\begin{align*}\Bigl(\Delta^3,\{\Delta^{\{2,3\}}\}, U_3\Bigr) \rightarrow \Bigl(\Delta^3,\{\Delta^{\{2,3\}}\}, \sharp \Bigr) \end{align*}$$

   where $U_3$ is the collections of all triangles except the $3$ -rd face.

9. (MSE) For every Kan complex K, the map

   $$\begin{align*}\Bigl( K,\flat,\sharp \Bigr) \rightarrow \Bigl(K,\sharp, \sharp\Bigr). \end{align*}$$

We will call a morphism in ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }} \mathbf {MS}$ -anodyne if it lies in the saturated hull of $\mathbf {MS}$ .

We can immediately obtain two useful lemmata.

Lemma 2.42. The morphism of marked simplicial sets

$$\begin{align*}\Bigl(\Delta^3,\flat, U_i\Bigr) \rightarrow \Bigl(\Delta^3,\flat, \sharp \Bigr), \quad 0<i<3, \end{align*}$$

where $U_i$ is the collection of all triangles except i-th face, is $\mathbf {MS}$ -anodyne.

Proof. See [25, Rmk 3.1.4].

Lemma 2.43. The morphism

$$\begin{align*}\theta\colon (\Delta^2,\{\Delta^{\{1,2\}},\Delta^{\{0,2\}}\},\sharp) \to (\Delta^2,\sharp,\sharp) \end{align*}$$

is $\mathbf {MS}$ -anodyne.

Proof. The proof follows exactly as in [5, Lem. 3.7].

Finally, in total analogy to the marked biscaled case, we can establish a pushout-product axiom, and thereby a model structure.

Proposition 2.44. Let $f:X\to Y$ be an $\mathbf {MS}$ -anodyne morphism in ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ , and let $g:A\to B$ be a cofibration in ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ . The morphism

$$\begin{align*}f\wedge g\colon X\times B\coprod_{X\times A} Y\times A \to Y\times B \end{align*}$$

is $\mathbf {MS}$ -anodyne.

Proof. Every case is, mutatis mutandis, the same as the corresponding case in the proof of [5, Prop. 3.10].

As in the marked-biscaled case, we can immediately define several mapping spaces.

Definition 2.45. Let $\overline {X}:=(X,E_X,T_X)$ be a fibrant marked-scaled simplicial set and $\overline {Y}:=(Y,E_Y,T_Y)$ any marked-scaled simplicial set. We can define a marked-scaled simplicial set ${\operatorname {Fun}}^{\mathbf {ms}}(\overline {Y},\overline {X})$ via the universal property

$$\begin{align*}{\operatorname{Hom}}_{{{\operatorname{Set}}^{\mathbf{ms}}_{\Delta}}}(\overline{A},{\operatorname{Fun}}^{\mathbf{ms}}(\overline{Y},\overline{X}))\cong {\operatorname{Hom}}_{{{\operatorname{Set}}^{\mathbf{ms}}_{\Delta}}}(\overline{A}\times \overline{Y},\overline{X}). \end{align*}$$

It follows from the pushout-product that this is a fibrant marked-scaled simplicial set, and thus that the underlying scaled simplicial set is an $\infty $ -bicategory. We denote this $\infty $ -bicategory by $\operatorname {Map}_{\mathbf {ms}}(\overline {Y},\overline {X})$ .

We can similarly define

• • A marked simplicial set $\operatorname {Map}_{\mathbf {ms}}^{\operatorname {th}}(\overline {Y},\overline {X})$ to be the full subsimplicial set of ${\operatorname {Fun}}^{\mathbf {ms}}(\overline {Y},\overline {X})$ consisting of the thin triangles.

• • A simplicial set $\operatorname {Map}_{\mathbf {ms}}^{\simeq }(\overline {Y},\overline {X})$ , which consists of precisely the marked edges in $\operatorname {Map}_{\mathbf {ms}}^{\operatorname {th}}(\overline {Y},\overline {X})$ .

Finally, we can establish the existence of the model structure:

Theorem 2.46. There is a left-proper combinatorial simplicial model category structure on ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ uniquely characterised by the following properties:

1. C) A morphism $f:X\to Y$ in ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ is a cofibration if and only if it is a monomorphism on underlying simplicial sets.

2. F) An object $X\in {{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ is fibrant if and only if the unique map $X\to \Delta ^0$ has the right lifting property with respect to the morphisms in $\mathbf {MS}$ .

Remark 2.47. It is not hard to see that we can tensor ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ over ${{\operatorname {Set}}_{\Delta }^+}$ and ${\operatorname {Set}}_\Delta $ in a way compatible with the enrichments provided by $\operatorname {Map}_{\mathbf {ms}}^{\operatorname {th}}(-,-)$ and $\operatorname {Map}_{\mathbf {ms}}^{\simeq }(-,-)$ , respectively. The latter of these provides the simplicial structure in the preceding proposition.

The weak equivalences in the model structure are precisely those $f:\overline {A}\to \overline {B}$ , which satisfy the equivalent conditions for any fibrant marked-scaled simplicial set $\overline {X}$ :

• • The induced map

  $$\begin{align*}\operatorname{Map}_{\mathbf{ms}}(\overline{B},\overline{X})\to \operatorname{Map}_{\mathbf{ms}}(\overline{A},\overline{X}) \end{align*}$$

  is a bicategorical equivalence.

• • The induced map

  $$\begin{align*}\operatorname{Map}_{\mathbf{ms}}^{\operatorname{th}}(\overline{B},\overline{X})\to \operatorname{Map}_{\mathbf{ms}}^{\operatorname{th}}(\overline{A},\overline{X}) \end{align*}$$

  is a weak equivalence of marked simplicial sets.

• • The induced map

  $$\begin{align*}\operatorname{Map}_{\mathbf{ms}}^{\simeq}(\overline{B},\overline{X})\to \operatorname{Map}_{\mathbf{ms}}^{\simeq}(\overline{A},\overline{X}) \end{align*}$$

  is a weak equivalence of Kan complexes.

It is not hard to see that the adjunction $D\dashv R$ can be promoted to a simplicial adjunction. By construction, L preserves cofibrations and R preserves fibrant objects, and thus we see that

Lemma 2.48. The adjunction

is a simplicial Quillen adjunction.

Further, we can define an adjunction

where $G(X,E_X,T_X)=(X,T_X)$ .

Lemma 2.49. The adjunction

is a Quillen adjunction.

Proof. It is immediate that $(-)^\flat $ preserves cofibrations. Suppose that $f:(X,T_X)\to (Y,T_Y)$ is a weak equivalence. Let $(Z,E_Z,T_Z)$ be a fibrant object in ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ . It is easy to see that $G(Z,E_Z,T_Z)=(Z,T_Z)$ is a fibrant object in ${\operatorname {Set}}_\Delta ^{\mathbf {sc}}$ . We can then note that, by definition, there is an isomorphism of mapping scaled simplicial sets

$$\begin{align*}\operatorname{Map}_{\mathbf{sc}}((X,T_X),(Z,T_Z))\cong \operatorname{Map}_{\mathbf{ms}}((X,\flat,T_X),(Z,E_Z,T_Z)). \end{align*}$$

Thus, since f induces a bicategorical equivalence

$$\begin{align*}\operatorname{Map}_{\mathbf{sc}}((Y,T_Y),(Z,T_Z))\to \operatorname{Map}_{\mathbf{sc}}((X,T_X),(Z,T_Z)) \end{align*}$$

we see that the map

$$\begin{align*}\operatorname{Map}_{\mathbf{ms}}((Y,\flat,T_Y),(Z,E_Z,T_Z))\to \operatorname{Map}_{\mathbf{ms}}((X,\flat,T_X),(Z,E_Z,T_Z)) \end{align*}$$

induced by $(f)^\flat $ is also an equivalence. We therefore see that $(f)^\flat $ is a weak equivalence in ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ , as desired.

Lemma 2.50. The functor G preserves weak equivalences.

Proof. If, for any $\infty $ -bicategory $(Z,T_Z)$ , there exists a set $E_Z$ of marked edges for Z such that $(Z,E_Z,T_Z)$ is a fibrant marked-scaled simplicial set, then this follows from the characterisation in terms of mapping $\infty $ -bicategories.

To see that this is the case, let $(Z,T_Z)$ be an $\infty $ -bicategory. Then $Z^{\operatorname {th}}$ is an $\infty $ -category, and so we can define a marking $E_Z$ on Z by declaring an edge to be marked if it lies in the maximal Kan complex in $Z^{\operatorname {th}}$ . From the definition, it is immediate that $(Z,E_Z,T_Z)$ has the extension property with respect to (MS1), (MS2),(MS3),(MS5),(MS6) and (MSE).

It follows from [5, Cor 4.20] and [5, Cor 4.23] that $Z\to \Delta ^0$ is a 2-Cartesian fibration in which the strongly Cartesian edges are precisely the equivalences, and so we see that $(Z,E_Z,T_Z)$ has the extension property with respect to (MS4), (MS7) and (MS8) as well.

Lemma 2.51. Given a fibrant marked-scaled simplicial set $(Y,E_Y,T_Y)$ , the full simplicial subset $Y^{\simeq }$ on the marked edges and scaled triangles is a Kan complex.

Proof. It is immediate from the definitions that $(Y^{\operatorname {th}},E_Y)$ is a fibrant marked simplicial set, and the lemma follows.

We now can state and prove the main proposition of this section.

Theorem 2.52. The Quillen adjunctions

and

are Quillen equivalences.

Proof. By [5, Thm 3.39], the composite adjunction $D\circ (-)^\flat \dashv G\circ R$ is a Quillen equivalence. It thus suffices for us to check that the adjunction $(-)^{\flat }\dashv G$ is a Quillen equivalence. We will check explicitly that the derived adjunction unit and counit are equivalences.

First, let $(X,T_X)\in {\operatorname {Set}}_\Delta ^{\mathbf {sc}}$ . The derived adjunction unit on $(X,T_X)$ is the composite

$$\begin{align*}(X,T_X) \to G(X,\flat,T_X) \to G((X,\flat,T_X)^{\operatorname{fib}}) \end{align*}$$

where the superscript $\operatorname {fib}$ denotes fibrant replacement. The first of these maps is the identity (since $G(X,\flat ,T_X)=(X,T_X)$ ) and the latter is the image under G of an equivalence of marked-scaled simplicial sets. By Lemma 2.50, this is an equivalence.

Now, let $(Y,E_Y,T_Y)\in {{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ be a fibrant object. The derived adjunction counit on $(Y,E_Y,T_Y)$ is the composite

Since every scaled simplicial set is cofibrant, the first map is an isomorphism, leaving us to check that the usual adjunction counit $\eta _Y$ is an equivalence. Note that $\eta _Y$ is simply the inclusion $(Y,\flat ,T_Y)\to (Y,E_Y,T_Y)$ .

We have a pushout square

and, by Lemma 2.51 the morphism $\psi $ is a morphism in $\mathbf {MS}$ of type (MSE). Thus, $\eta _Y$ is $\mathbf {MS}$ -anodyne, and is a weak equivalence.

2.4.1. The ${\operatorname {Set}}_\Delta ^+$ -enrichment on ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$

We have already constructed a model structure on the category ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ of marked-scaled simplicial sets, and shown that it is a simplicial model category with respect to the mapping spaces ${\operatorname {Map}}^{\simeq }_{\mathbf {ms}}(-,-)$ . However, we will need to consider ${\operatorname {Set}}_\Delta ^+$ -enriched functors in our analysis of the Grothendieck construction. Our aim in this section is therefore to show that our model structure can, additionally, be viewed as ${{\operatorname {Set}}_{\Delta }^+}$ -enriched. The following lemma constitutes an easy first check in this direction.

Lemma 2.53. The category ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ is powered and tensored over ${\operatorname {Set}}_\Delta ^+$ via the maps

$$\begin{align*}{\operatorname{Set}}_\Delta^+\times {\operatorname{Set}}_\Delta^{\mathbf{ms}}\to {\operatorname{Set}}_\Delta^{\mathbf{ms}}, \enspace (K,X) \mapsto K_\sharp \times X \end{align*}$$

and

$$\begin{align*}{[-,-]}\colon {\operatorname{Set}}_\Delta^+\times {\operatorname{Set}}_\Delta^{\mathbf{ms}} \to {\operatorname{Set}}_\Delta^{\mathbf{ms}},\enspace (K,X) \mapsto {\operatorname{Fun}}^{\mathbf{ms}}(K_\sharp,X) \end{align*}$$

The tensoring and powering is compatible with the mapping spaces ${\operatorname {Map}}^{\operatorname {th}}_{\mathbf {ms}}(-,-)$ .

Our aim throughout the rest of the section will be to show that the tensoring is a left Quillen bifunctor. We will follow the strategy of [10], showing first that the model structure on ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ is a Cisinski-Olschok model structure (as with ${\operatorname {Set}}_\Delta ^{\mathbf {sc}}$ in [11]), and then using testing pushout-products with the concomitant interval objects. For a review of Cisinski-Olschok model structures, see the appendix of [11].

We first show that the model structure on ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ is Cartesian-closed. This will follow immediately from Proposition 2.44 and the following

Lemma 2.54. Let $f:X\to Y$ and $g:A\to B$ be two weak equivalences in ${\operatorname {Set}}_\Delta ^{\mathbf {mb}}$ , then the product

$$\begin{align*}f\times g\colon X\times A \to Y\times B \end{align*}$$

is a weak equivalence.

Proof. Precisely the same argument as in [25, Lemma 4.2.6] allows us to reduce to the case of the morphism

$$\begin{align*}Y\times A\to Y\times B \end{align*}$$

where Y, A and B are all fibrant objects. By the characterisation of fibrant objects, this morphism is a weak equivalence if and only if the morphism on underlying scaled simplicial sets is an equivalence, which follows from loc. cit.

Corollary 2.55. For any cofibrations $f:X\to Y$ and $g:A\to B$ , the pushout-product

$$\begin{align*}f\wedge g\colon Y\times A \coprod_{X\times A} X\times B \to Y\times B \end{align*}$$

is an equivalence if one of f or g is.

Proof. We can use the small object argument to factor f as

where h is MS-anodyne. Consequently, k is a weak equivalence. We consider the diagram

It follows from the lemma that the bottom horizontal arrow is a weak equivalence, and the top horizontal arrow is the induced map on homotopy colimits by a natural weak equivalence. From Proposition 2.44, it follows that the right-hand morphism is an equivalence, and the corollary follows.

Corollary 2.56. The model structure on ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ is Cartesian-closed.

We now wish to show that the Cisinski-Olschok model structure on ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ with interval $\Delta ^0\amalg \Delta ^0 \to (\Delta ^1)^\sharp _\sharp $ and generating anodyne maps the MS-anodyne maps is, in fact the model structure constructed in our previous section. We first note that, since one of the morphism $\Delta ^0\to (\Delta ^1)^\sharp _\sharp $ is MS-anodyne, it follows that both such morphisms are trivial cofibrations.

Definition 2.57. We write $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {CO}}$ for the Cisinski-Olschok model structure on ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ with interval $\Delta ^0\amalg \Delta ^0\to (\Delta ^1)^\sharp _\sharp $ , and generating set of anodyne morphisms the set of MB-anodyne morphisms.

For ease, we will write $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {AH}}$ for the model structure previously defined

Proposition 2.58. The two model structures $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {CO}}$ and $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {AH}}$ coincide.

Proof. It will suffice to show that the fibrant objects coincide. By construction, every fibrant object of $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {CO}}$ is a fibrant object of $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {AH}}$ . However, since $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {AH}}$ is a Cartesian-closed model category, and the interval object for $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {CO}}$ is a cylinder in $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {AH}}$ , every anodyne map in $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {CO}}$ is a trivial cofibration in $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {AH}}$ . Thus every fibrant object of $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {AH}}$ is a fibrant object of $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})_{\operatorname {CO}}$ .

As a consequence, we will now drop the unwieldy subscript notation for the model structure on ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ . We can now prove the following.

Proposition 2.59. The model category ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ is a ${\operatorname {Set}}_\Delta ^+$ -enriched model category.

Proof. We need only show that the tensoring satisfies the pushout-product axiom, i.e., that for cofibrations $f:K\to S$ in ${{\operatorname {Set}}_{\Delta }^+}$ and $g:X\to Y$ in ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ , the pushout-product $f\wedge g$ is a trivial cofibration that either f or g is. Since both model structures are Cisinski-Olschok model structures, it suffices to test generating monomorphisms against the two interval inclusions and against the generating anodyne morphisms.

It is immediate from Proposition 2.44 that if f (resp. g) is marked (resp. MS) anodyne, then $f\wedge g$ is a trivial cofibration. It remains for us to test the cases when f is $\{0\}\to (\Delta ^1)^\sharp $ or $\{1\}\to (\Delta ^1)^\sharp $ , and the cases when g is $\{0\}\to (\Delta ^1)^\sharp _\sharp $ or $\{1\}\to (\Delta ^1)^\sharp _\sharp $ .

However, since the morphisms $\{0\}\to (\Delta ^1)^\sharp _\sharp $ or $\{1\}\to (\Delta ^1)^\sharp _\sharp $ are trivial cofibrations, and the model structure on ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ is Cartesian-closed, this follows immediately.

3. The bicategorical Grothendieck construction

Our first step towards an $\infty $ -bicategorical Grothendieck construction is defining the functors which will realise the desired equivalence. These definitions will constitute an upgrade of the straightening and unstraightening constructions of [23, Section 3.2] to the more highly decorated setting of marked-biscaled simplicial sets and marked-scaled simplicial sets. These functors will define a Quillen equivalence of model categories between $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ and a model category we now define.

Definition 3.1. Let be a ${\operatorname {Set}}_{\Delta }^+$ -category. We denote by the category of ${\operatorname {Set}}^{+}_{\Delta }$ -enriched functors and natural transformations. We endow the category of enriched functors with the projective model structure (See, e.g., [23, A.3.3.2]).

Definition 3.2. Let $(Y,E_Y,T_Y)$ be a marked scaled simplicial set. We define a scaled simplicial set which we denote $(Y^{\triangleright },T_{Y^{\triangleright }})$ whose underlying simplicial set is given by $Y^{\triangleright }=Y \ast \Delta ^0$ and whose non-degenerate thin simplices are either those that factor through Y or those of the form $f \ast \operatorname {id}_{\Delta ^0}$ where $f:\Delta ^1 \to Y$ belongs to $E_Y$ .

Remark 3.3 (Important convention).

Let $(X,M_X,T_X\subseteq C_X)$ be an $\textbf {MB}$ simplicial set. By the underlying scaled simplicial set, we will mean the scaled simplicial set $(X,T_X)$ .

Remark 3.4 (Notation for ${\operatorname {op}}$ s).

Given a simplicial set X with any decoration (marking, scaling, etc.), we will denote by $X^{\operatorname {op}}$ the opposite simplicial set with the same decoration.

Given an enriched category (a ${\operatorname {Set}}_\Delta ^+$ -enriched category, a 2-category, etc.), we will denote the enriched category with the same objects and . In the specific case of a 2-category $\mathbb {C}$ , we will occasionally write ${\mathbb {C}}^{({\operatorname {op}},-)}$ to denote ${\mathbb {C}}^{\operatorname {op}}$ . We will only rarely make use of the 2-morphism dual ${\mathbb {C}}^{(-,{\operatorname {op}})}$ .

We now provide the underlying left Quillen functor of our bicategorical Grothendieck construction.

Construction 3.5. Fix a scaled simplicial set $S\in {{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ and a functor of ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories . Let $p:X\to S$ be an object of $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ . We define a scaled simplicial set $X_S$ via the pushout diagram

We generically denote both the cone point of $X^\triangleright $ and its image in $X_S$ by $\ast $ . We then define a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched category

Note that this is equivalently the pushout

of ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories.

Applying the enriched Yoneda embedding on the cone point $\ast $ , this provides a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor

We promote this functor to a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor

by equipping its values on objects with a scaling.

After a single, fairly ad-hoc definition, we are able to do this in a highly functorial way. The ad-hoc definition will be a promotion of ${\mathfrak {C}}^{{\operatorname {sc}}}[X^\triangleright ]$ to a ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ -enriched category, such that the subcategory ${\mathfrak {C}}^{{\operatorname {sc}}}[X]\subset {\mathfrak {C}}^{{\operatorname {sc}}}[X^\triangleright ]$ has all mapping spaces maximally scaled. We will denote the resulting ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ -enriched category ${\mathfrak {C}}^{\operatorname {sc}}[X^\triangleright ]_\dagger $ . More generally, we will denote scalings on the mapping spaces of a marked-simplicially enriched category using subscripts, e.g. for maximally marked mapping spaces.

We will define the scaling on ${\mathfrak {C}}^{{\operatorname {sc}}}[X^\triangleright ]$ in three steps:

1. 1. We define the scaling

   $$\begin{align*}{\mathfrak{C}}^{\operatorname{sc}}[X^\triangleright]_\dagger(s,t):= {\mathfrak{C}}^{\operatorname{sc}}[X^\triangleright](s,t)_\sharp \end{align*}$$

   for $s,t\in X$ .

2. 2. We define an auxiliary scaling $P_{X^\triangleright }^s$ on each marked simplicial set ${\mathfrak {C}}^{{\operatorname {sc}}}[X^\triangleright ](s,\ast )$ . Given a map $\sigma \colon \Delta ^n\to X$ , we can pass to the associated $n+1$ -simplex $\sigma \star \operatorname {id}_0:\Delta ^{n+1}\to X^\triangleright $ and obtain a map of simplicial sets

   $$\begin{align*}{\mathfrak{C}}^{{\operatorname{sc}}}[\Delta^{n+1}](0,n+1)\to {\mathfrak{C}}^{{\operatorname{sc}}}[X^{\triangleright}](\sigma(0),\ast). \end{align*}$$

   Each 2-simplex in ${\mathfrak {C}}^{{\operatorname {sc}}}[\Delta ^{n+1}](0,n+1)$ is of the form

   $$\begin{align*}S_0\cup \{n+1\}\subset S_1\cup \{n+1\}\subset S_2 \cup \{n+1\} \end{align*}$$

   where $S_i\subseteq [n]$ contains $0$ . We declare the image of such a 2-simplex to be scaled in ${\mathfrak {C}}^{\operatorname {sc}}[X^\triangleright ](s,\ast )$ precisely when either

   • • $\max (S_i)=\max (S_j)$ for some $i,j\in \{0,1,2\}$ ; or

   • • the simplex $\sigma $ is a lean 2-simplex in X (i.e., lies in $C_X$ ) and the 2-simplex is $03\to 013\to 0123$ .

   The auxiliary scaling $P^s_{X^\triangleright }$ then consists of all such 2-simplices.

3. 3. We extend the scaling $P^s_{X^\triangleright }$ by functoriality. That is, we declare a 2-simplex $\sigma :\Delta ^2\to \mathfrak {C}[X^\triangleright ](s,\ast )$ to be scaled if there is a t in X and a 2-simplex

   $$\begin{align*}\theta=(\theta_1,\theta_2)\colon \Delta^2 \to {\mathfrak{C}}^{\operatorname{sc}}[X^\triangleright](s,t) \times {\mathfrak{C}}^{\operatorname{sc}}[X^\triangleright](t,\ast) \end{align*}$$
   such that $\theta _2\circ \theta _1=\sigma $ , where $\theta _2\in P^s_{X^\triangleright }$ . We would like to stress to the reader that this also adds scaled 2-simplices in the case where $\theta _2$ is degenerate.

We can then define a ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ -enriched variant of $X_\phi $ to be the pushout of ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ -enriched categories

Unwinding the definitions, we see that a 2-simplex $\sigma :\Delta ^2\to \overline {X_\phi }(s,\ast )$ is scaled if and only if it satisfies the following condition:

• • There is a $t\in X_\phi $ and a 2-simplex

  $$\begin{align*}\theta=(\theta_1,\theta_2)\colon \Delta^2 \to \overline{X_\phi}(s,t) \times \overline{X_\phi}(t,\ast) \end{align*}$$

  such that (1) $\sigma =\theta _2\circ \theta _1$ , and, (2) $\theta _2$ is either in the image of an element of $P^s_{X^\triangleright }$ or is degenerate.

The bicategorical straightening of X is then the restriction of the ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ -enriched Yoneda embedding:

A priori, this is an ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ -enriched functor. However, since we required the mapping spaces in to be maximally scaled, this formula in fact defines an ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor. This construction then yields a functor

which we call the (bicategorical) straightening functor.

Notation. We will denote by $\mathbb {S}\!{\operatorname {t}}_S(X)$ the special case in which $\phi :{\mathfrak {C}}^{{\operatorname {sc}}}[S]\to {\mathfrak {C}}^{{\operatorname {sc}}}[S]$ is the identity.

Remark 3.6. In line with the philosophy of [32], there should be a model for $(\infty ,3)$ -categories on the category of simplicial sets with decorations on 1-, 2-, and 3-simplices. The ad-hoc construction of the ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ -enriched category $\overline {X_\phi }$ above seems likely to fit into some – as-yet-undefined – $(\infty ,3)$ -categorical version of the rigidification functor, which turns decorated 3-simplices in scaled 2-simplices in the corresponding mapping space.

Remark 3.7. Given a 2-Cartesian fibration $p:X\to S$ , we note that if every triangle in X is lean, the map $\mathbb {S}\!{\operatorname {t}}_S(X)(i)\to \mathbb {S}\!{\operatorname {t}}_S(X)(i)_\sharp $ is an equivalence of marked-scaled simplicial sets. More generally, we obtain a diagram

which commutes up to natural weak equivalence.

While we will not formalise this statement here, there should be a model structure on $({{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }})_{/S}$ modeling $\infty $ -bicategories fibred in $(\infty ,1)$ -categories, such that $\operatorname {St}^+$ becomes a left Quillen equivalence to the projective model structure. The diagram above would then represent the restriction of our straightening-unstraightening equivalence to this special case.

3.1. First properties

Before proceeding to the technical nitty-gritty of the Quillen equivalences, we establish some basic properties of the straightening functor.

Proposition 3.8. Let $S\in {{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ and let be a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor. Then the following hold

1. 1. The straightening functor $\mathbb {S}\!{\operatorname {t}}_\phi $ preserves colimits.

2. 2. (Base change for scaled functors) Given a morphism of scaled simplicial sets $f: T\to S$ there is a diagram

   

   which commutes up to natural isomorphism of functors.

3. 3. (Base change for ${{\operatorname {Set}}_{\Delta }^+}$ -functors) Given a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor there is a diagram

   

   which commutes up to natural isomorphism of functors.

Proof. All three statements hold on the level of $\operatorname {St}_\phi ^+$ , and so the proof amounts to checking scalings. We prove (1), and leave the other two statements to the reader.

It follows from the definition that

preserves colimits. Since colimits in functor categories are computed pointwise, it will thus suffice to show that, given a diagram

$$\begin{align*}D \colon I\to ({\operatorname{Set}}_\Delta^{\mathbf{mb}})_{/S} \end{align*}$$

the scalings on $\operatorname *{\mathrm {colim}}_I {\mathbb {S}\text {t}}_\phi (D(i))$ and ${\mathbb {S}\text {t}}_\phi (\operatorname *{\mathrm {colim}}_I D(i))$ coincide. Indeed, applying the universal property, it will suffice to show that the map

$$\begin{align*}{\mathbb{S}\text{t}}_\phi(\operatorname*{\mathrm{colim}}_I D(i)) \to \operatorname*{\mathrm{colim}}_I {\mathbb{S}\text{t}}_\phi(D(i)) \end{align*}$$

which is the identity on underlying marked simplicial sets preserves the scalings.

Fix , we will first show that the map

$$\begin{align*}f_s\colon ({\mathbb{S}\text{t}}_\phi(\operatorname*{\mathrm{colim}}_I D(i)))(s) \to (\operatorname*{\mathrm{colim}}_I {\mathbb{S}\text{t}}_\phi(D(i)))(s) \end{align*}$$

preserves the scalings inherited from $P_{X^\triangleright }^s$ . To this end, suppose given a thin simplex $\sigma $ in $P_{({\mathbb {S}\text {t}}_\phi (\operatorname *{\mathrm {colim}}_I D(i)))_S}^s$ which does not come from a lean simplex in the colimit. Tracing through the definition, we note that there must be a simplex $\eta :\Delta ^n\to \operatorname *{\mathrm {colim}}_I(D(i))$ and a simplex $\mu :=\{S_0\cup \{n+1\}\to S_1\cup \{n+1\}\to S_2\cup \{n+1\}\}$ in ${\mathfrak {C}}^{\operatorname {sc}}[\Delta ^{n+1}](0,n+1)$ with $\max (S_i)=\max (S_j)$ for some $i,j=0,1,2$ such that $\sigma $ is the image of $\mu $ under the canonical map

$$\begin{align*}g_1\colon{\mathfrak{C}}^{\operatorname{sc}}[\Delta^{n+1}](0,n+1)\to ({\mathbb{S}\text{t}}_\phi(\operatorname*{\mathrm{colim}}_I D(i)))(s) \end{align*}$$

is not scaled.

By the construction of colimits in simplicial sets, this means that there is a $k\in I$ and a simplex $\hat {\eta }:\Delta ^n\to D(k)$ such that $\eta $ factors through the canonical map $D(k)\to \operatorname *{\mathrm {colim}}_I D(i)$ as $\hat {\eta }$ . We can then note that $\hat {\eta }$ will yield a map

$$\begin{align*}g_2\colon{\mathfrak{C}}^{\operatorname{sc}}[\Delta^{n+1}](0,n+1)\to {\mathbb{S}\text{t}}_\phi(D(k))(s)\to (\operatorname*{\mathrm{colim}}_I{\mathbb{S}\text{t}}_\phi(D(i)))(s) \end{align*}$$

such that the diagram

commutes. We thus see that $g_2(\mu )=f_s(g_1(\mu ))=f_s(\sigma )$ is scaled, as desired. The same argument holds, mutatis mutandis, for $\sigma \in P_{({\mathbb {S}\text {t}}_\phi (\operatorname *{\mathrm {colim}}_I D(i)))_S}^s$ coming from a lean 2-simplex in the colimit.

We can now easily check that the full scalings $T_{X_S}^s$ are preserved by $f_s$ by simply noting that the diagram

commutes.

To show (2) and (3), we again note that the statements are immediate if we replace ${\mathbb {S}\text {t}}_\phi $ with $\operatorname {St}_\phi ^+$ (cf. [25, Rmk 3.5.16] and [23, Prop 3.2.1.4]). A similar check to the above assures us that the scalings coincide.

Remark 3.9. Note that, in the case where we consider $\phi $ to be the identity on ${\mathfrak {C}}^{{\operatorname {sc}}}[S]$ and are given a morphism $f:T\to S$ , combining (2) and (3) in Proposition 3.8 yields a diagram

which commutes up to natural isomorphism.

Corollary 3.10. Let S be a scaled simplicial set and a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor. Then the straightening functor $\mathbb {S}\!{\operatorname {t}}_\phi $ has a right adjoint

which we call the (bicategorical) unstraightening functor.

Proof. This follows from the first part in Proposition 3.8 using the adjoint functor theorem.

Let $\Delta ^n_{\flat }$ denote the minimally scaled n-simplex and consider $(\Delta ^n)^{\flat }_{\flat }=(\Delta ^n,\flat ,\flat )$ as an object of $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/\Delta ^n_{\flat }}$ via the identity map. To ease the notation we will denote the straightening of this object as $\mathbb {S}\!{\operatorname {t}}_{\Delta ^n_\flat }\!(\Delta ^n)$ .

Definition 3.11. Let $n\geqslant 0$ and consider a $\textbf {MB}$ simplicial set $\Delta ^n_{T}:=(\Delta ^n,\flat ,\flat \subseteq T)$ for some lean-scaling T. Given $0\leqslant s \leqslant n$ we define a marked-scaled simplicial set $\mathcal {L}^n_T(s)$ as follows:

• • The underlying simplicial set is given (by the nerve) of the poset $L^n(s)$ of subsets $S \subseteq [n]$ such that $\min (S)=s$ ordered by inclusion.

• • Let $\sigma :S_0 \subseteq S_1 \subseteq S_2$ be a 2-simplex in (the nerve of) $L^n(s)$ and denote $s_i=\max (S_i)$ for $i=0,1,2$ . We declare $\sigma $ to be thin if the simplex $\{s_0,s_1,s_2\}$ is lean in $\Delta ^n_T$ .

• • An edge in $\mathcal {L}^n_T(s)$ is declared marked if and only if it is degenerate.

Lemma 3.12. Let $\Delta ^n_{T}:=(\Delta ^n,\flat ,\flat \subseteq T)$ and denote by $\mathbb {S}\!{\operatorname {t}}_{\Delta ^n_\flat }\! (\Delta ^n_T)$ the straightening of the map $\Delta ^n_{T} \to \Delta ^n_\flat $ . Then for every $0\leqslant s \leqslant n$ the canonical map

$$\begin{align*}\mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}\!( \Delta^n_T)(s) \to \mathcal{L}^n_{T}(s) \end{align*}$$

is $\mathbf {MS}$ -anodyne.

Proof. The existence of the morphism is clear from the definitions. Moreover, if T is given by the minimal lean-scaling the map is in fact an isomorphism.

Suppose that we are given a thin 2-simplex $\sigma :S_0 \subseteq S_1 \subseteq S_2$ in $\mathcal {L}^n_{T}(i)$ . As before, we adopt the convention that $s_i:=\max (S_i)$ . We will show that $\sigma $ can be scaled by taking pushouts along $\mathbf {MS}$ -anodyne morphisms. First let us consider the 3-simplex

We immediately observe that all of its faces are scaled in $\mathbb {S}\!{\operatorname {t}}_{\Delta ^n_\flat }\!( \Delta ^n)_T(s)$ except the face missing . It follows we can scale the remaining face using a pushout along a $\mathbf {MS}$ -anodyne map of the type described in Lemma 2.42. Now we consider another 3-simplex

Again we observe that all of its faces are scaled except possibly the face missing which is precisely $\sigma $ . The conclusion easily follows from Lemma 2.42

Let $S_{\sharp }$ be a scaled simplicial set and assume every triangle is thin. Denote by S its underlying simplicial set and let $({\operatorname {Set}}_{\Delta }^{+})_{/S}$ denote the category of marked simplicial sets over S. We define a functor

$$\begin{align*}\iota\colon({\operatorname{Set}}_{\Delta}^{+})_{/S} \to ({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/S}; (X,E_X) \mapsto (X,E_X, \sharp) \end{align*}$$

We view the $\infty $ -categorical straightening functor $\operatorname {St}_S$ (see 3.2.1 in [23]) as a functor with values $\left ({\operatorname {Set}}^{\mathbf {ms}}_{\Delta }\right )^{{\mathfrak {C}}^{\operatorname {sc}}[S]^{\operatorname {op}}}$ by maximally scaling the values of $\operatorname {St}_S\!X(s)$ .

Proposition 3.13. There exists a natural transformation

$$\begin{align*}\varepsilon\colon\mathbb{S}\!{\operatorname{t}}_{S}\circ \iota \Rightarrow \operatorname{St}_S \end{align*}$$

which is objectwise a weak equivalence of marked scaled simplicial sets.

Proof. The existence of the natural transformation is automatic since both functors only differ on the scaling. It is clear that both functors preserve colimits and that they satisfy base change with respect to morphisms of simplicial sets $S \to T$ (see Proposition 3.8). In addition, it is routine to verify that both functors respect cofibrations. A standard argument then shows that it suffices to check that the natural transformation is an equivalence (1) when $S=(\Delta ^n)^{\flat }$ with $n\geqslant 0$ and $X \to S$ is the identity morphism, and (2) on $(\Delta ^1)^{\sharp } \to \Delta ^1$ when $S=\Delta ^1$ . This is a direct consequence of Lemma 3.12.

We conclude this section with a first step towards showing that the bicategorical straightening is left Quillen.

Proposition 3.14. Let S be a scaled simplicial set and let $\phi :{\mathfrak {C}}^{{\operatorname {sc}}}[S]\to \mathcal {C}$ be a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor. Then the straightening functor

$$\begin{align*}\mathbb{S}\!{\operatorname{t}}_{\phi}\!\colon({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/S} \to \left({\operatorname{Set}}^{\mathbf{ms}}_{\Delta}\right)^{\mathfrak{C}[\mathcal{C}]^{\operatorname{op}}} \end{align*}$$

preserves cofibrations.

Proof. The generators of the class of cofibrations of marked biscaled simplicial sets are given by

1. (C1) $\left (\partial \Delta ^n,\flat ,\flat \right ) \rightarrow \left ( \Delta ^n,\flat ,\flat \right )$ .

2. (C2) $\left (\Delta ^1,\flat ,\flat \right ) \rightarrow \left (\Delta ^1,\sharp ,\flat \right ) $ .

3. (C3) $\left (\Delta ^2,\flat ,\flat \right ) \rightarrow \left (\Delta ^2,\flat ,\flat \subset \sharp \right )$ .

4. (C4) $\left (\Delta ^2,\flat ,\flat \subset \sharp \right ) \rightarrow \left ( \Delta ^2,\flat ,\sharp \right )$ .

Note that (C4) and (S2) are the same morphism. Therefore using standard arguments it will suffice to check our claim on those generators.

Let $i:A \to B$ be a cofibration. As stated above it will suffice to check in the case where i is one of the generating cofibrations. Furthermore we can use Proposition 3.8 to reduce to the case where S is the underlying scaled simplicial set of B, and $\phi $ is $\operatorname {id}:{\mathfrak {C}}^{{\operatorname {sc}}}[S]\to {\mathfrak {C}}^{{\operatorname {sc}}}[S]$ .

Let be a trivial fibration in $\left ({\operatorname {Set}}^{\mathbf {ms}}_{\Delta }\right )^{\mathfrak {C}[\mathcal {C}]^{\operatorname {op}}}$ . Then in particular for every $c \in \mathcal {C}$ the map $F(c) \to G(c)$ is a trivial fibration of marked-scaled simplicial sets and thus detects scalings and markings. This settles the cases (C2)-(C4). For the remaining case, we observe that

$$\begin{align*}{\mathbb{S}\text{t}}_{\Delta^n_\flat}(\partial\Delta^n)(i) \to {\mathbb{S}\text{t}}_{\Delta^n_\flat}(\Delta^n)(i) \end{align*}$$

is an isomorphism unless $i=0$ , in which case the map is a cofibration. The claim follows immediately.

3.2. Products and tensoring

Before we can proceed to proving that the straightening-unstraightening adjunction is a Quillen equivalence (indeed, before we can prove the straightening is left Quillen), we need to establish the relation of the straightening to the ${{\operatorname {Set}}_{\Delta }^+}$ -tensoring. We will prove this as a corollary of a more general result – on products of MB simplicial sets – which will be of use to us in the sequel.

Let $A,B \in {\operatorname {Set}}_\Delta ^{\mathbf {sc}}$ and consider a pair of objects $X_A \in ({\operatorname {Set}}^{\mathbf {mb}}_\Delta )_{/A}$ , $X_B \in ({\operatorname {Set}}^{\mathbf {mb}}_\Delta )_{/B}$ giving rise to $X=X_A\times X_B \in ({\operatorname {Set}}^{\mathbf {mb}}_\Delta )_{/A\times B}$ then we can form a pushout diagram

where the left-most vertical morphism is the composite $ {\mathfrak {C}}^{\operatorname {sc}}[X] \to {\mathfrak {C}}^{\operatorname {sc}}[A\times B] \to {\mathfrak {C}}^{\operatorname {sc}}[A] \times {\mathfrak {C}}^{\operatorname {sc}}[B]$ . Let

$$\begin{align*}{\mathbb{S}\text{t}}_A X_A \boxtimes {\mathbb{S}\text{t}}_B X_B: {\mathfrak{C}}^{\operatorname{sc}}[A]^{{\operatorname{op}}} \times {\mathfrak{C}}^{\operatorname{sc}}[B]^{{\operatorname{op}}} \to {\operatorname{Set}}^{\mathbf{ms}}_\Delta \end{align*}$$

be the pointwise product of ${\mathbb {S}\text {t}}_A X_A$ and ${\mathbb {S}\text {t}}_B X_B $ and observe there is a canonical natural transformation

We will prove the following theorem:

Theorem 3.15. The map is a pointwise weak equivalence.

Before proceeding with the proof of the theorem we need to do some preliminary work. First we will do a careful study of the case where $A=(\Delta ^n,\flat )$ and $B=(\Delta ^k,\flat )$ , $X_A=(\Delta ^n,\flat ,\flat )$ and $X_B=(\Delta ^k,\flat ,\flat )$ . We will assume that the maps $X_A \to A$ and $X_B \to B$ are the identity on the underlying scaled simplicial sets. In this particular situation we will denote ${\mathbb {S}\text {t}}_{\phi }X(i,j):=\mathbb {P}^{n,k}_{(i,j)}$ and ${\mathbb {S}\text {t}}_{\Delta ^n}\Delta ^n(i) \times {\mathbb {S}\text {t}}_{\Delta ^k}\Delta ^k(j):=\mathbb {S}^{n,k}_{(i,j)}$ .

Definition 3.16. Let $n,k \geqslant 0$ and let $i \in [n], j \in [k]$ . We define marked scaled simplicial set $\mathbb {E}^{n,k}_{(i,j)}$ whose underlying simplicial set is given by $\mathfrak {C}[(\Delta ^n \times \Delta ^k)^{\triangleright }]((i,j),\ast )$ . To define the marking and the scaling we construct a morphism

$$\begin{align*}\xi^{n,k}_{(i,j)}\colon\mathbb{E}^{n,k}_{(i,j)} \to \mathbb{S}^{n,k}_{(i,j)} \end{align*}$$

and equip $\mathbb {E}^{n,k}_{(i,j)}$ with the induced marking and scaling. Recall that objects of $\mathbb {E}^{n,k}_{(i,j)}$ are given by a chain or sequence of inequalities $(a_0,b_0)<(a_1,b_1)< \cdots (a_\ell ,b_\ell )$ where $a_i \in [n]$ and $b_i \in [k]$ for $i=0,\dots ,\ell $ and with the property that $(a_0,b_0)=(i,j)$ . We will use the notation . A morphism between chains $C_1 \to C_2$ is simply given by an inclusion $C_1 \subset C_2$ which we call a refinement of the chain $C_1$ . Then we define $\xi (C)=(S_a,S_b)$ where and similarly for $S_b$ .

Remark 3.17. It is immediate to see that the map $\xi ^{n,k}_{(i,j)}$ constructed before factors as

$$\begin{align*}\mathbb{E}^{n,k}_{(i,j)} \to \mathbb{P}^{n,k}_{(i,j)} \to \mathbb{S}^{n,k}_{(i,j)} \end{align*}$$

where the second morphism is the component of the natural transformation $\varepsilon _X$ at the object $(i,j)$ and the first morphism is a canonical collapse map. We will denote the first morphism by $\pi ^{n,k}_{(i,j)}$ and the second morphism by $\varepsilon ^{n,k}_{(i,j)}$ .

Definition 3.18. Let $C \in \mathbb {E}^{n,k}_{(i,j)}$ be a chain denoted by . We set $|C|=\ell $ and we call it the length of the chain.

Definition 3.19. Let $C \in \mathbb {E}^{n,k}_{(0,0)}$ . We define $\mathcal {E}_{C}$ to be the full subposet (with the induced marking and scaling) of $\mathbb {E}^{n,k}_{(0,0)}$ consisting of those chains K contained in C.

Definition 3.20. Let $C \in \mathbb {E}^{n,k}_{(0,0)}$ be a chain. We say that $K \in \mathcal {E}_{C}$ is a rigid chain if there is no marked morphism in $\mathcal {E}_{C}$ with source K. We denote the by $\mathcal {E}_{C}^{r}$ the full subposet of $\mathcal {E}_{C}$ on rigid chains.

Lemma 3.21. Let $C \in \mathbb {E}^{n,k}_{(0,0)}$ be a chain and denote by $\mathcal {U}_C$ the image of the morphism $\mathcal {E}_{C} \to \mathbb {S}^{n,k}_{(0,0)}$ . Then $\xi ^{n,k}_{(0,0)}$ induces an isomorphism of marked scaled simplicial sets

Proof. The map $\xi ^r_C$ is clearly surjective on vertices. Moreover, given a morphism $U\to K$ in $\mathcal {E}_C$ , choose a marked morphism $U\to U^r$ to a rigid chain in $\mathcal {E}_C$ . Then for every , the object $K\cup \{(a,b)\}$ will lie in $\mathcal {E}_C$ over the same element of $\mathcal {U}_C$ as K. We thus obtain a morphism $U^r\to \hat {K}$ lying over the original morphism in $\mathcal {U}_C$ , showing that $\xi ^r_C$ is surjective on morphisms, and thus on higher simplices.

Moreover $\xi ^r_C$ detects and preserves marked edges and thin simplices. It will therefore suffice to show that $\xi _C^r$ is injective. Let $K_i \in \mathcal {E}^r_C$ for $i=1,2$ such that $\xi ^r_C(K_1)=\xi ^r_C(K_2)$ . Let us denote for $i=1,2$ . Without loss of generality let us assume that we have some $(a^1_s,b^1_s)$ such that this pair is not an element in $K_2$ . However, note that since $K_i \subset C$ for $i=1,2$ then there exists a map $K_2 \to \hat {K}_2$ where $\hat {K}_2$ is obtained from $K_2$ by appending the element $(a^1_s,b^1_s)$ . By construction it follows that $\xi ^r_C(K_2)=\xi ^r_C(\hat {K}_2)$ since $K_2$ is rigid it follows that $\hat {K}_2=K_2$ and therefore $K_1=K_2$ .

Lemma 3.22. Let $C \in \mathbb {E}^{n,k}_{(0,0)}$ . Then the induced morphism

is an equivalence of marked scaled simplicial sets.

Proof. Let $\iota :\mathcal {E}_C^r \to \mathcal {E}_C$ denote the obvious inclusion. Using Lemma 3.21 we can construct a map $s_C=\iota \circ (\xi ^r_C)^{-1}$ . It is clear that $\xi _C \circ s_C=\operatorname {id}$ . Given K, observe that by construction $s_C\circ \xi _C(K)$ is rigid. Let $K \to K^r$ be a marked edge where $K^r$ is rigid. Since the restriction of $\xi _C$ to rigid objects is injective it follows that $s_C \circ \xi _C(K)= K^r$ . This yields a marked homotopy from $s_C \circ \xi _C$ to the identity and the result follows.

Lemma 3.23. Let $C_i \in \mathbb {E}^{n,k}_{(0,0)}$ for $i=1,2$ . Then there exists a chain K such that the intersection $\mathcal {E}_{C_1} \cap \mathcal {E}_{C_2}=\mathcal {E}_K$ .

Proof. Immediate.

Proposition 3.24. Let $n,k$ two non-negative integers and consider $i \in [n]$ and $j \in [k]$ . Then the morphism

$$\begin{align*}\xi^{n,k}_{(i,j)}\colon\mathbb{E}^{n,k}_{(i,j)} \to \mathbb{S}^{n,k}_{(i,j)} \end{align*}$$

is an equivalence of marked scaled simplicial sets.

Proof. First let us observe that the map $\xi ^{n,k}_{(i,j)}$ is an isomorphism if either n or k is equal to $0$ . Using an inductive argument it will suffice to show that the map $\xi ^{n,k}_{(0,0)}$ is an equivalence. Note that we can cover $\mathbb {E}^{n,k}_{(i,j)}$ with the subsimplicial sets $\mathcal {E}_C$ where C is a chain of maximal length. Since $\xi ^{n,k}_{(0,0)}$ is surjective, $\mathbb {S}^{n,k}_{(0,0)}$ is covered by the subsimplicial sets $\mathcal {U}_C$ . Applying [4, Lemma 3.2.13], we express $\mathbb {E}^{n,k}_{(0,0)}$ and $\mathbb {S}^{n,k}_{(0,0)}$ as the colimit over the same diagram of two homotopy cofibrant diagrams. We can now identify $\xi ^{n,k}_{(0,0)}$ as the map induced by the natural transformation whose components are $\xi _C$ . Therefore using Lemma 3.22 it follows that $\xi ^{n,k}_{(0,0)}$ is a weak equivalence.

Proof of Theorem 3.15.

Let us commence the proof by observing that since the morphism ${\mathfrak {C}}^{\operatorname {sc}}[A \times B] \to {\mathfrak {C}}^{\operatorname {sc}}[A] \times {\mathfrak {C}}^{\operatorname {sc}}[B]$ is a weak equivalence we can use [23, Proposition A.3.3.8.] and (3) in Proposition 3.8 to reduce our problem to showing that the map

is a pointwise weak equivalence. As in [23, 3.2.1.13], it suffices to check this in the special case when $X_A\to A$ and $X_B\to B$ are identity morphisms on underlying simplicial sets, and both A and B are one of the following cases

• • The scaled 2-simplex $\Delta ^2_\sharp $ .

• • The unscaled n-simplex $\Delta ^n_\flat $ .

In the case where $A=\Delta ^n_\flat $ and $B=\Delta ^k_\flat $ , the morphism

$$\begin{align*}\mathbb{S}\!{\operatorname{t}}_{A \times B}(\Delta^n\times \Delta^k)(i,j) \to \phi^*\left(\mathbb{S}\!{\operatorname{t}}_{\Delta^n}(\Delta^n)\boxtimes \mathbb{S}\!{\operatorname{t}}_{\Delta^k}(\Delta^k)\right)(i,j) \end{align*}$$

is precisely the morphism

$$\begin{align*}\xi^{n,k}_{(i,j)}\colon\mathbb{E}^{n,k}_{(i,j)} \to \mathbb{S}^{n,k}_{(i,j)} \end{align*}$$

and thus is an equivalence of marked-scaled simplicial sets by Proposition 3.24. Each other case is a pushout of some $\xi _{(i,j)}^{n,k}$ by a cofibration, and thus is also an equivalence.

3.3. Straightening and anodyne morphisms

This section serves as a stepping-stone to see that the bicategorical straightening is a left Quillen functor. In particular, we will show that $\mathbb {S}\!{\operatorname {t}}_S$ preserves MB-anodyne morphisms for any $S\in {{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ .

Definition 3.25. Consider $\Lambda ^n_i$ for $0 \leqslant i \leqslant n$ . For every $0\leqslant s \leqslant n$ we define $\Lambda \mathcal {L}^n_i (s)$ to be the scaled subsimplicial set of $\mathcal {L}^n_\flat (s)$ consisting of those simplices $\sigma : S_0 \subseteq S_1 \subseteq \cdots \subseteq S_\ell $ satisfying at least one of the following conditions:

1. i) There exists $k\in [n]$ with $k\neq i$ such that, for every $t\in [\ell ]$ , $k\notin S_t$ .

2. ii) There exists some $0<k\leqslant n$ such that $k \in S_0$ and there exists $0\leqslant u < k$ such that $u \neq i$ .

Given $\Delta ^n_{T}$ as in Definition 3.11 we define $(\Lambda \mathcal {L}^n_i)_{T}(s)$ using the inherited scaling from $\mathcal {L}^n_{T}(s)$ .

Definition 3.26. Given a MB simplicial set of the form $\Delta ^n_T:= (\Delta ^n,\flat ,\flat \subset T)$ for some T, we denote by $(\Lambda ^n_i)_T$ the horn with the induced marking and biscaling. We write $\mathbb {S}\!{\operatorname {t}}_{\Delta ^n_\flat }\!(\Lambda ^n_i)_T$ for the functor associated to the object $(\Lambda ^n_i)_{T} \to \Delta ^n_\flat $ .

Remark 3.27. In some specific instances we will have $\Delta ^n_T:= (\Delta ^n,\flat ,\flat \subset T)$ where a chosen 2-simplex in $\Delta ^n$ . In that situation we will chose a subscript notation . This convention will also applied to previously defined constructions like for example $(\Lambda ^n_i)_\dagger $ or $\mathbb {S}\!{\operatorname {t}}_{\Delta ^n}\!(\Delta ^n)_\dagger $ .

Lemma 3.28. Let $\Delta ^n_{T}=(\Delta ^n,\flat ,\flat \subseteq T)$ . Then for every $0 \leqslant s \leqslant n$ the canonical morphism

is $\mathbf {MS}$ -anodyne.

Proof. It is clear that for every $0 \leqslant s \leqslant n$ the map in the statement is an isomorphism on the underlying simplicial sets since its just the restriction of the isomorphism described at the beginning of Lemma 3.12 to the corresponding horn. We further note that the proof of Lemma 3.12 still holds in this setting. Consequently, the claim follows.

Definition 3.29. Let $n\geqslant 0$ and $0 \leqslant s \leqslant n$ . We say that a (non-degenerate) simplex $\sigma $ in $\mathcal {L}^n(s)$ is a path if it is of maximal dimension. Let $\mathcal {P}^n_s$ be the set of such paths. We will define an total order on $\mathcal {P}^n_s$ as follows:

Given a path $\sigma :S_0 \subset S_1 \subset S_2 \subset \cdots S_\ell $ one sees that consists precisely in one element. Therefore we can identify $\sigma $ with a list of elements

$$\begin{align*}S_\sigma=\{ a_{i}\}_{i=1}^{\ell}. \end{align*}$$

Note that by the maximality of $\sigma $ , $S_0=\{s\}$ .

Suppose we are given two such lists $S_\sigma =\{ a_{i}\}_{i=1}^{\ell }$ and $S_\theta =\{ b_{i}\}_{i=1}^{\ell }$ . We declare $\sigma < \theta $ if for the first index j for which $a_{j} \neq b_{j}$ then we have $a_{j} < b_{j} $ .

Lemma 3.30. Let and consider the induced morphism $(\Lambda \mathcal {L}^n_0)_{\diamond }(0) \to \mathcal {L}^n_{\diamond }(0)$ . Collapsing the morphism $0 \to 01$ to a degenerate edge on both sides yields a map of scaled simplicial sets

$$\begin{align*}\Lambda\mathcal{R}^n_0 =(\Lambda \mathcal{L}^n_0)_{\diamond}(0) \coprod_{\Delta^{1}}\Delta^{0} \to \mathcal{L}^n_{\diamond}(0) \coprod_{\Delta^{1}}\Delta^{0}=\mathcal{R}^n \end{align*}$$

which is scaled anodyne.

Proof. We use the order from Definition 3.29 to add simplices to $\Lambda \mathcal {R}^n_0$ . We will add simplices in reverse order, i.e. for any path $\sigma $ , we denote by $X^{\geqslant \sigma }$ the scaled simplicial subset of $\mathcal {R}^n$ obtained by adding to $\Lambda \mathcal {R}^n_0$ all paths $\theta $ such that $\theta \geqslant \sigma $ .

The procedure yields a filtration

$$\begin{align*}\Lambda \mathcal{R}^n_0=X^{\geqslant \sigma_0} \to X^{\geqslant \sigma_1} \to X^{\geqslant \sigma_2} \to \cdots \to \mathcal{R}^n \end{align*}$$

where we have labeled our paths $\sigma _i$ so that $\sigma _i>\sigma _{i+1}$ . The proof proceeds by showing that $X^{\geqslant \sigma _{i-1}}\to X^{\geqslant \sigma _{i}}$ is scaled anodyne for any i.

The proof proceeds by cases. We fix the notation that $S_{\sigma _i}=\{a_k\}_{k=1}^n$ .

1. 1. Suppose that $a_1\neq 1$ . We prove this case by showing that the top horizontal map in the pullback diagram

   

   is itself scaled anodyne.

   We see that $A_{\sigma _i}$ is the union of the following faces of $\sigma _i$ :

   1. a) The face $d_0(\sigma _i)$ , since we will have $0\leqslant 1<a_1$ in each $S_k$ (see ii) in Definition 3.25).

   2. b) The face $d_n(\sigma _i)$ , since this face will always be missing $a_{n}\neq 0$ (see i) in Definition 3.25).

   3. c) The face $d_j(\sigma _i)$ for every j such that $a_{j+1}>a_j$ . This is because this will, equivalently, be the j ^th face of the (greater) simplex with vertex list

      $$\begin{align*}\{a_1,\ldots, a_{j-1},a_{j+1},a_j,a_{j+2},\ldots, a_n\}. \end{align*}$$

   To see that the inclusion of $A_{\sigma _i}\to \Delta ^n$ is scaled anodyne, we first note that, by necessity, there is at least one face of $\Delta ^n$ not contained in $A_{\sigma _i}$ . We then choose the biggest $t\in [n]$ such that $d_t(\Delta ^n)$ is not contained in $A_{\sigma _i}$ and note that $t<n$ . We wish to apply Lemma 2.36 with $\mathcal {A}$ given by the subsets of the form $\{0\},\{n\}$ and $\{j\}$ as in c) above, and where the pivot point is t. Note that by construction $\{t+1\} \in \mathcal {A}$ . We conclude after noting that $\Delta ^{\{s,t,t+1\}}$ is scaled for every $\{s\} \in \mathcal {A}$ , since $\max (S_t)=\max (S_{t+1})$ by maximality of t. Consequently, we see that $\alpha _i$ is scaled anodyne, as desired.

2. 2. Now suppose that $a_1=1$ , and $S_{\sigma _i}\neq \{1,2,\ldots ,n-1,n\}$ . We now must instead consider the pullback diagram

   
   as above, we see that $B_{\sigma _i}$ consists of the faces

   1. a) $d_n(\sigma _i)$

   2. b) $d_j(\sigma _i)$ for each j such that $a_{j+1}>a_j$ .

   Since $S_{\sigma _i}\neq \{1,2,\ldots ,n-1,n\}$ , there exists some $1<t<n$ such that $B_{\sigma _i}$ does not contain the t ^th face of $\Delta ^n\coprod _{\Delta ^{\{0,1\}}}\Delta ^0$ . We can apply again Lemma 2.36 with pivot point given the biggest t satisfying the conditions above and where $\mathcal {A}$ is given by $\{n\}$ and $\{j\}$ for elements j as in b).

3. 3. If $S_{\sigma _i}=\{1,2,\ldots ,n-1,n\}$ , then the map $X^{\geqslant \sigma _{i-1}}\to X^{\geqslant \sigma _i}=\mathcal {R}^n$ is obtained as a pushout along the inclusion

   $$\begin{align*}\Lambda^n_0\coprod_{\Delta^{\{0,1\}}}\Delta^0 \to \Delta^n\coprod_{\Delta^{\{0,1\}}}\Delta^0 \end{align*}$$
   where $\Delta ^{\{0,1,n\}}$ is scaled.

Lemma 3.31. Let and consider the induced morphism $(\Lambda \mathcal {L}^n_n)_{\dagger }(0) \to \mathcal {L}^n_{\dagger }(0)$ . Denote by $\mathcal {T}^n$ (resp. $\Lambda \mathcal {T}^n_n$ ) the marked scaled simplicial set obtained from $\mathcal {L}^n_n(0)$ (resp. $(\Lambda \mathcal {L}^n_n)_{\dagger }(0)$ ) by marking the edges $S \to S \cup \{n\}$ where $\max (S)=n-1$ . Then the associated map

$$\begin{align*}\Lambda \mathcal{T}^n_n \to \mathcal{T}^n \end{align*}$$

is $\mathbf {MS}$ -anodyne.

Proof. The argument is nearly identical to the proof of Lemma 3.30. Using the same order as in that proof, we produce a filtration

$$\begin{align*}\Lambda \mathcal{T}^n_n=X^{\geqslant \sigma_{0}} \to X^{\geqslant \sigma_{1}}\to \cdots \to \mathcal{T}^n \end{align*}$$

and show each step is scaled anodyne.

As before, we set $S_{\sigma _i}=\{a_j\}_{j=1^n}$ , and consider the pullback diagram

The case distinction now rests on whether or not $d_n\sigma _i$ factors through $A_{\sigma _i}$ . The case when it does is formally identical to case (1) from Lemma 3.30.

If $d_n(\sigma _i)$ does not factor through $A_{\sigma _i}$ , then $a_n=n$ . There are again two cases, based on whether $S_{\sigma _i}=\{1,2\ldots ,n-1,n\}$ . The case $S_{\sigma _i}\neq \{1,2\ldots ,n-1,n\}$ is identical to the corresponding case in Lemma 3.30. In the case $S_{\sigma _i}=\{1,2\ldots ,n-1,n\}$ we find that $A_{\sigma _i}=\Lambda ^n_n$ , the last edge is marked, and $\Delta ^{\{0,n-1,n\}}$ is scaled. This is a morphism of type 2.41, and thus is $\mathbf {MS}$ -anodyne.

Lemma 3.32. Let where $0<i<n$ and consider the induced morphism $(\Lambda \mathcal {L}^n_i)_{\ast _i}(0) \to \mathcal {L}_{\ast _{i}}(0)$ . Let $\mathcal {S}_i^n$ (resp $\Lambda \mathcal {S}^n_i$ ) denote the marked scaled simplicial set obtained by marking the edges of the form $S \to S'$ such that $ i+1 \in S$ but $i \notin S$ and such that . Then the induced morphism

$$\begin{align*}\Lambda \mathcal{S}^n_i \to \mathcal{S}^n_i \end{align*}$$

is $\mathbf {MS}$ -anodyne.

Proof. Let $S_\tau =\{1,2,\ldots ,i-1,i+1,\ldots ,n-1,n,i\}$ and denote by $\hat {\tau }$ the smallest maximal simplex such that $\hat {\tau }>\tau $ . We define a filtration as in Lemma 3.30 and Lemma 3.31 up until the stage $X^{\geqslant \hat {\tau }}$ , yielding

$$\begin{align*}\Lambda\mathcal{S}^n_i \to X^{\geqslant \sigma_1} \to \cdots \to X^{\geqslant \hat{\tau}} \to \mathcal{S}^n_i. \end{align*}$$

We will first prove that that each step of this factorisation is $\mathbf {MS}$ -anodyne, making a distinction into 2 cases.

We consider the map $X^{\geqslant \sigma _{k-1}}\to X^{\geqslant \sigma _k}$ , and set $S_{\sigma _k}:=\{a_j\}_{j=1}^n$ . We again form the pullback

We then have two cases.

1. 1. If $S_{\sigma _k}$ has as its last entry anything other than i, then $A_{\sigma _k}$ consists of

   • • The face $d_0(\sigma _k)$ .

   • • The face $d_n(\sigma _k)$ .

   • • The face $d_j(\sigma _k)$ for each j such that $a_{j+1}> a_j$ .

   The argument is then nearly identical to case (1) from Lemma 3.30.

2. 2. If the last entry of $S_{\sigma _k}$ is i, then $A_{\sigma _k}$ consists of

   • • The face $d_0(\sigma _k)$ .

   • • The face $d_j(\sigma _k)$ for each j such that $a_{j+1}>a_j$ .

   The remainder of the argument is nearly identical to case (2) of Lemma 3.30.

It now remains only for us to show that is $\mathbf {MS}$ -anodyne. For ease of notation, we set $Z:=X^{\geqslant \hat {\tau }}$ .

We now need to add the remaining simplices. Write $\Sigma ^\leqslant $ for the set of maximal simplices which are less than or equal to $\tau $ . Given $\theta \in \Sigma ^{\leqslant }$ , we write $S_\theta =\{b_j\}_{j=1}^n$ for the ordered vertex sequence, as usual. We further denote by $\widehat {S_\theta }$ the vertex sequence obtained by removing i. We will call a simplex $\theta \in \Sigma ^{\leqslant }$ disordered if $\widehat {S_\theta }>\widehat {S_{\tau }}$ . If $\widehat {S_\theta }=\widehat {S_{\tau }}$ , we will call $\theta $ calm.³

Our first order of business is to add the disordered simplices in $\Sigma ^{\leqslant }$ . For each disordered $\theta $ , we define $Z^{\geqslant \theta }$ to be obtained from Z by adding all the disordered simplices $\sigma $ for which $\sigma \geqslant \theta $ . Applying the order induced on disordered simplices, we again obtain a filtration

$$\begin{align*}Z\to Z^{\geqslant \sigma_1} \to Z^{\geqslant \sigma_2}\to \cdots \to Z^{\geqslant \gamma} \end{align*}$$

where $\gamma $ is the minimal disordered simplex under the order $<$ .

As before, we form a pullback diagram

and show that $\beta _k$ is $\mathbf {MS}$ -anodyne. Note that $B_{\sigma _k}$ consists precisely of

• • The face $d_0(\sigma _k)$ .

• • The face $d_n(\sigma _k)$ (since the final entry of $S_{\sigma _k}$ cannot be i).

• • The face $d_j(\sigma _k)$ for each j such that $a_{j+1}>a_j$ .

The argument that $\beta _k$ is anodyne is, by now, routine.

We now turn to adding the calm simplices. Notice that $\tau $ is the maximal calm simplex. We now set $Y:=Z^{\geqslant \gamma }$ , and define a filtration

by defining $Y^{\leqslant \theta }$ to be the union of Y with all of the calm simplices less than or equal to $\theta $ .

For every calm $\sigma _k$ other than $\tau $ itself, and with corresponding sequence $S_{\sigma _k}=\{a_j\}_{j=1}^{n}$ , we obtain a pullback diagram

where $a_\ell =i$ and $0<\ell <n$ . If $S_{\sigma _k}=\{1,2,3,\ldots , n-1,n\}$ , then this is a $\Lambda ^n_i$ , and the scaling on $\Delta ^n_{\ast _i}$ shows us that the simplex $\Delta ^{\{i-1,i,i+1\}}\subset \sigma _k$ is scaled. On the other hand, if $S_{\sigma _k}\neq \{1,2,3,\ldots , n-1,n\}$ , the simplex $\Delta ^{\{\ell -1,\ell ,\ell +1\}}\subset \Lambda ^n_\ell $ is already scaled in . The morphism $\eta _k$ is thus a scaled anodyne map, and the pushout is therefore $\mathbf {MS}$ -anodyne.

We are left only to add $\tau $ . However, in this case, we obtain a pullback diagram

where the 2-simplex $\Delta ^{\{0,n-1,n\}}$ is scaled and the edge $\Delta ^{\{n-1,n\}}$ is marked. The result then follows from a pushout of type (MS4).

Proposition 3.33. Let S be a scaled simplicial set and let ${\mathfrak {C}}^{{\operatorname {sc}}}[S]\to \mathcal {C}$ be a functor of ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories. Consider an MB-anodyne morphism $i:A \to B$ in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ . Then for every $s \in S$ then induced map

$$\begin{align*}\mathbb{S}\!{\operatorname{t}}_{\phi}\! A(s) \to \mathbb{S}\!{\operatorname{t}}_{\phi}\!B(s) \end{align*}$$

is a trivial cofibration of marked scaled simplicial sets.

Proof. As in the proof of Proposition 3.14, we can assume that $S=B$ , that $\phi $ is $\operatorname {id}:{\mathfrak {C}}^{{\operatorname {sc}}}[S]\to {\mathfrak {C}}^{{\operatorname {sc}}}[S]$ , and that i is one of the generators in Definition 2.18. We proceed to verify each case.

1. A1) It is immediate that ${\mathbb {S}\text {t}}_B A(s)\to {\mathbb {S}\text {t}}_B B(s)$ is an isomorphism when $s\neq 0$ . Lemma 3.32 shows that the map is $\mathbf {MS}$ -anodyne when $s=0$ .

2. A2) Note that the morphism $\mathbb {S}\!{\operatorname {t}}_{B}\!A(s) \to \mathbb {S}\!{\operatorname {t}}_B\!B(s)$ is an isomorphism for $s \neq 0$ . If $s=0$ the map is an isomorphism on the underlying marked simplicial sets. Let (see Definition 2.18) and let $\mathcal {L}^4_{T}(0)$ and $\mathcal {L}^4_{\hat {T}}$ be the simplicial sets defined in Definition 3.11 equipped with the marking given by the thin simplices in the base. We obtain a commutative diagram

   

   where the vertical morphisms are equivalences due to Lemma 3.12. We will show that the bottom morphism is an equivalence. Observe that once we manage to scale the simplices $0\to 01 \to 014$ and $0 \to 03 \to 034$ then rest of the scaling follows using the argument given in Lemma 3.12. We start by considering the 4-simplex given by

   $$\begin{align*}0 \to 01 \to 012 \to 0123 \to 01234 \end{align*}$$

   and note that by taking a pushout along a morphism of type 2.41 we can scale the simplices and . Now we consider a 3-simplex

   $$\begin{align*}0 \to 01 \to 014 \to 01234 \end{align*}$$

   where all of its faces are now scaled except possibly the 3rd face. We further note that we can factor the last morphism as $014 \to 0134 \to 01234$ where both morphisms are marked. Therefore we can assume without loss of generality that the map $014 \to 01234$ is also marked. This allows us to scale the 3rd face using a pushout along a map of type 2.41. Inspecting the 3-simplex

   $$\begin{align*}0 \to 03 \to 0123 \to 01234 \end{align*}$$

   we see that we can add to the scaling by Lemma 2.42. Finally let us consider

   $$\begin{align*}0 \to 03 \to 034 \to 01234. \end{align*}$$

   As we did before we factor the last map as a composite of marked morphisms $034 \to 0134 \to 01234$ . The claim follows by a totally analogous argument as before.

3. A3) Let $\ast $ denote the vertex to which $0$ and $1$ get identified. Then it follows that the induced map $\mathbb {S}\!{\operatorname {t}}_{B}\!A(s) \to \mathbb {S}\!{\operatorname {t}}_B\!B(s)$ is an isomorphism for $s \neq \ast $ . Lemma 3.30 shows that the map is $\mathbf {MS}$ -anodyne when $s=0$ .

4. A4) It is immediate that $\mathbb {S}\!{\operatorname {t}}_{B}\!A(s) \to \mathbb {S}\!{\operatorname {t}}_B\!B(s)$ is an isomorphism for $s \neq 0$ . Lemma 3.31 shows that the map is $\mathbf {MS}$ -anodyne when $s=0$ .

5. S2) The induced map is an isomorphism for every object of $\Delta ^2$ .

6. S3) The map is an isomorphism for every $s \in \Delta ^3$ such that $s \neq 0$ . We will prove the case $i=1$ leaving the case $i=2$ as an exercise to the reader. Let $\mathcal {L}^3_{U_1}(0)$ and $\mathcal {L}^3_{\sharp }(0)$ be as in Definition 3.11 and equip them with the marking induced by the thin simplex . We obtain a commutative diagram

   

   where the vertical morphisms are equivalences due to Lemma 3.12. Therefore it will enough to show that the bottom morphism is an anodyne map of marked scaled simplicial sets. Consider the simplex $0 \to 01 \to 012 \to 0123$ and observe that all of its faces are scaled except the face missing $01$ . Therefore we can scale the $01$ -face using a pushout along an anodyne morphism as described in Lemma 2.42. Now we consider $0 \to 02 \to 012 \to 0123$ and we observe that we can scale the face missing $012$ by another pushout. Finally we look at $0 \to 02 \to 023 \to 0123$ and we note that the last edge must be marked and that all of the faces are scaled except the face missing the vertex $0123$ . Thefore another pushout along a morphism of type (MS8) yields the result.

7. S4) & S5) The proof is very similar to the previous case and left to the reader.

8. A5,S1 & E) Since these maps are always maximally thin scaled we can use Proposition 3.13 and apply the pertinent proofs in Proposition 3.2.1.11 in [23].

3.4. Straightening over the point

In this section, we will prove two important results. We will show that the the bicategorical straightening functor is left Quillen over any scaled simplicial set, and we will show that the straightening is an equivalence over the point. We fix the notation ${\mathbb {S}\text {t}}_{\Delta ^0}={\mathbb {S}\text {t}}_*$ .

Definition 3.34. We define an adjunction

where $L(X,E_X,T_X \subseteq C_X):=(X,E_X,C_X)$ and $R(Y,E_Y,T_Y)=(Y,E_Y,T_Y \subseteq T_Y)$ . We note that $L\circ R=\operatorname {id}$ and that the unit natural transformation $(X,E_X,T_X \subseteq C_X) \to (X,E_X,C_X \subseteq C_X)$ is MB-anodyne. It is easy to see that L preserves cofibrations and trivial cofibrations. In particular, we see that $L \dashv R$ is a Quillen equivalence.

Our goal in this section is to construct a natural transformation which is a levelwise weak equivalence. By general abstract nonsense, it will suffice to construct morphisms $ \alpha _X: \mathbb {S}\!{\operatorname {t}}_*(X) \to L(X)$ whenever X is one of the following generators

• • $\Delta ^n_{\flat }:=(\Delta ^n,\flat ,\flat )$ , for $n\geqslant 0$ ,

• • $\Delta ^2_{\dagger }:=(\Delta ^2,\flat ,\flat \subset \Delta ^2)$ ,

• • $\Delta ^2_{\sharp }:=(\Delta ^2,\flat ,\Delta ^2)$ ,

• • $(\Delta ^1)^\sharp :=(\Delta ^1,\Delta ^1,\flat )$ ,

and to prove that that the maps $\alpha _X$ are natural with respect to morphisms among generators. The next step is to give a precise description of the straightening functor applied to those generators.

Definition 3.35. Let $n\geqslant 0$ and define a simplicial set

$$\begin{align*}Q^n:=\bigsqcup\limits_{0 \leqslant i \leqslant n}{\mathbb{O}}^{n+1}(i,n+1)_{\big/ \sim} \end{align*}$$

where the relation $\sim $ identifies simplices $\sigma _1 \in {\mathbb {O}}^{n+1}(i,n+1)$ and $\sigma _2 \in {\mathbb {O}}^{n+1}(j,n+1)$ with $i \leqslant j$ whenever $\sigma _1$ is in the image of the map

We further observe that the morphisms

assemble into a map $\alpha _n:Q^n \to \Delta ^n$ . We wish now to upgrade $Q^n$ to a scaled simplicial set. We do so by declaring a triangle $\sigma :\Delta ^2 \to Q^n$ thin if and only if its image under $\alpha _n$ is degenerate in $\Delta ^n$ . We denote this collection of thin triangles by $T_{Q^n}$ .

Remark 3.36. Given an order preserving morphism $f:[n] \to [m]$ then it is straightforward to check that we can produce a commutative diagram

Lemma 3.37. We have the following isomorphisms of marked scaled simplicial sets

• • $\mathbb {S}\!{\operatorname {t}}_*(\Delta ^n_{\flat }) \simeq (Q^n,\flat ,T_{Q^n})$ .

• • ${\mathbb {S}\text {t}}_*(\Delta ^2_{\dagger })={\mathbb {S}\text {t}}_*(\Delta ^2_{\sharp }) \simeq (Q^2,\flat ,\sharp )$ .

• • ${\mathbb {S}\text {t}}_*((\Delta ^1)^{\sharp })=(Q^1,\sharp ,\flat )$ .

Lemma 3.38. The morphism

$$\begin{align*}\alpha_n\colon Q^n \to \Delta^n_{\flat} \end{align*}$$

is a weak equivalence of marked scaled simplicial sets.

Proof. We construct a section $s:\Delta ^n_{\flat } \to Q^n$ by sending $i \in [n]$ to the set and note that $\alpha _n \circ s=\operatorname {id}_{\Delta ^n}$ . To finish the proof we will construct a marked homotopy between $\operatorname {id}_{Q^n}$ and $s \circ \alpha _n$ .

Let $\sigma :\Delta ^k \to Q^n$ and pick a representative $S_0 \subseteq S_1 \subseteq \cdots \subseteq S_k$ with $S_j \in {\mathbb {O}}^{n+1}(i,n+1)$ for $0 \leqslant j \leqslant k$ . To ease the notation we will omit the element $n+1$ from the subsets $S_j$ . Let us denote $s_j=\max (S_j)$ and observe that we can produce a diagram $H(\mathord {-},\sigma ):\Delta ^1 \times \Delta ^k \to Q^n$

It is straightforward to check that if $\sigma \sim \theta $ then $H(\mathord {-},\sigma ) = H(\mathord {-},\theta )$ . We have now constructed a natural transformation $\Delta ^1 \times Q^n \to Q^n$ . It is immediate to see that $H(0,\mathord {-})=\operatorname {id}_{Q^n}$ . In addition the fact that the bottom row in the diagram is equivalent to

ensures that $H(1,\mathord {-})=s \circ \alpha _n$ . We conclude the proof by noting that the morphism $S_j \to [i,s_j]$ gets collapsed to a degenerate edge and that all squares in the homotopy are thin. Thus we have produced a marked homotopy and so the result follows.

Proposition 3.39. There exists a natural transformation which is a levelwise weak equivalence.

Proof. Using Lemma 3.37 is immediate to verify that the maps (together with decorated variants) $\alpha _n: Q^n \to \Delta ^n$ assemble into a natural transformation . To check that $\alpha $ is a levelwise equivalence, we note that due to the fact that both ${\mathbb {S}\text {t}}_*$ and L are left adjoints which preserve cofibrations it will suffice to check on generators. We proceed case by case

• • $\alpha _{n}:(Q^n,\flat ,T_{Q^n}) \to (\Delta ^n,\flat ,\flat )$ is an equivalence due to Lemma 3.38.

• • $\alpha _2^{\sharp }:(Q^2,\flat ,\sharp ) \to (\Delta ^2,\flat ,\sharp )$ is an equivalence since we can repeat the proof above with maximally scaled simplicial sets.

• • $\alpha _1^{\sharp }:(Q^1,\sharp ,\flat ) \to (\Delta ^1,\sharp ,\flat )$ is an isomorphism.

Theorem 3.40. Let S be a scaled simplicial set, then the bicategorical straightening functor

$$\begin{align*}\mathbb{S}\!{\operatorname{t}}_S\!\colon({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/S} \to \left({\operatorname{Set}}^{\mathbf{ms}}_{\Delta}\right)^{{\mathfrak{C}}^{\operatorname{sc}}[S]^{\operatorname{op}}} \end{align*}$$

is a left Quillen functor.

Proof. Given a weak equivalence $f\colon X\to Y$ in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ , we can apply fibrant replacement to obtain a commutative diagram

where the horizontal morphisms are MB-anodyne, and there vertical morphisms are weak equivalences.

Since $\mathbb {S}\!{\operatorname {t}}_S$ preserves MB-andoyne morphisms, we may thus assume without loss of generality that X and Y are fibrant objects. By [5, Lemma 3.29], f then has a homotopy inverse g. Let

$$\begin{align*}H\colon(\Delta^1)^\sharp_\sharp \times X\to X \end{align*}$$

be a marked homotopy between $g\circ f$ and $\operatorname {id}_X$ over S. Then $\mathbb {S}\!{\operatorname {t}}_S(H)$ factors as

Where $\varepsilon $ is an equivalence by Theorem 3.15, $\alpha $ is an equivalence by Proposition 3.39, and the final map is an equivalence since $(\Delta ^1)^\sharp \to \Delta ^0$ is an equivalence of marked simplicial sets. Since $\mathbb {S}\!{\operatorname {t}}_S(g\circ f)$ and $\mathbb {S}\!{\operatorname {t}}_S(\operatorname {id}_X)=\operatorname {id}_{\mathbb {S}\!{\operatorname {t}}_S(X)}$ are both sections of $\mathbb {S}\!{\operatorname {t}}_S(H)$ , they are thus equivalent in the homotopy category. An identical argument shows that $\mathbb {S}\!{\operatorname {t}}_S(f\circ g)\simeq \operatorname {id}_{\mathbb {S}\!{\operatorname {t}}_S(Y)}$ , completing the proof.

Corollary 3.41. In particular the adjunction

is a Quillen equivalence.

Proof. By Proposition 3.39, $\operatorname {St}_\ast $ is naturally equivalent to a left Quillen equivalence. The corollary follows immediately.

3.5. Straightening over a simplex

As in [25, Ch. 2], the proof that our Grothendieck construction is a Quillen equivalence over a general scaled simplicial set will be bootstrapped from a proof over the n-simplices $(\Delta ^n)_\flat $ . In analogy to the method in op. cit., we will prove this case by constructing a mapping simplex for each 2-Cartesian fibration $X\to \Delta ^n_\flat $ – a tractable $\textbf {MB}$ simplicial set $\mathcal {M}_X\to \Delta ^n_\flat $ which is equivalent to X over $\Delta ^n_\flat $ .⁴ The majority of this section is given over to showing that we can decompose a 2-Cartesian fibration $X\to \Delta ^n_\flat $ as a homotopy pushout of the restriction of X to $\Delta ^{n-1}$ , which enables the inductive step of our proof.

Remark 3.42. The term ‘mapping simplex’ used above is potentially misleading. In [23] and [25], a mapping simplex is a fibration over $\Delta ^n$ explicitly constructed from a functor or a . Our construction makes use of no such functor, and thus is not a true mapping simplex in this sense. The abuse of the term mapping simplex in the above exposition should be seen as suggestive of the role this construction fills in our proof – one roughly analogous to the role of the mapping simplex in the proof of the $(\infty ,1)$ -categorical Grothendieck construction in [23, §3.2].

Definition 3.43. We define a marked biscaled simplicial set $(\Delta ^n)^{\diamond }:=(\Delta ^n,E^n_{\diamond },\flat \subset \sharp )$ where $E^n_{\diamond }$ is the collection of all edges containing the vertex n. It is not hard to verify that the inclusion of the terminal vertex is MB-anodyne.

For the rest of this section, we fix a 2-Cartesian fibration $p:X \to \Delta ^n$ over the minimally scaled n-simplex. We consider the commutative diagram

where $X_n$ denotes the fibre over the vertex n and the dotted arrow exists due to the fact that the left vertical morphism is MB-anodyne.

Consider the inclusion morphism $\iota :\Delta ^{[0,n-1]} \to \Delta ^n$ an equip $\Delta ^{[0,n-1]}$ with the structure of an $\textbf {MB}$ simplicial set by declaring and edge (resp. triangle) marked (resp. thin, resp. lean) if its image in $\Delta ^n$ is marked (resp. thin, resp. lean) in $(\Delta ^n)^{\diamond }$ . Notice that this amounts to equipping $\Delta ^{n-1}$ with the minimal marking and thin-scaling, and the maximal lean-scaling.

We denote the restriction of X to $\Delta ^{n-1}$ by $X|_{\Delta ^{n-1}}:=X\times _{\Delta ^{n}}\Delta ^{n-1}$ , and denote the restriction of $\alpha $ to $X_n\times (\Delta ^{n-1})^\diamond $ by $\alpha ^\prime $ . We construct an $\textbf {MB}$ simplicial set $\mathcal {M}_X$ over $\Delta ^n$ by means of the pushout square

Note that, since the top horizontal map is a cofibration, this is a homotopy pushout square in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/\Delta ^n_\flat }$ . The morphism $\alpha $ and the inclusion $X|_{\Delta ^{n-1}}\to X$ yield a cone over this diagram, and thus a canonical morphism $\omega :\mathcal {M}_X \to X$ over $\Delta ^n$ . The key technical element in this section will be to show that $\omega $ is a weak equivalence in the 2-Cartesian model structure.

Definition 3.44. Let $\sigma :\Delta ^k \to X$ . Given $I\subset [k]$ , we define

Given $J \subset I \subseteq [k]$ and $\theta \in F_I(\sigma )$ such that $\theta \neq \ast $ we denote by $d_{J,I}(\theta )$ the image of $\theta $ in $\mathcal {M}_X$ under the face operator induced by the inclusion $J \subset I$ .

Definition 3.45. We define a MB simplicial set $\mathcal {L}_X$ whose simplices $\sigma :\Delta ^k \to \mathcal {L}_X$ are given by:

• • A simplex $\hat {\sigma }:\Delta ^k \to X$ .

• • For every non-empty subset $I\subseteq [k]$ an element $\theta _I \in F_I(\sigma )$ . If $\theta _I=*$ we use the empty set notation $\theta _I=\varnothing $ .

We impose to this data the following compatibility conditions

1. H1) Given $J \subset I \subseteq [k]$ and $\theta _I \in F_I(\sigma )$ such that $\theta _I \neq \varnothing $ it follows that $d_{J,I}(\theta _I)=\theta _J$ .

2. H2) Given $I \subseteq [k]$ with $i_m=\max (I)$ , then if $p \circ \hat {\sigma }(i_m) \neq n$ it follows that $\theta _I \neq \varnothing $ .

3. H3) Given $I \subseteq [k]$ such that for every $i \in I$ we have $p \circ \hat {\sigma }(i)=n $ , then it follows that $\theta _I \neq \varnothing $ .

Notice that by construction there is a canonical projection map $v:\mathcal {L}_X \to X$ . We equip $\mathcal {L}_X$ with the marking and biscaling induced by v.

Given a simplex $\sigma :\Delta ^k\to \mathcal {L}_X$ , we refer to the collection $\{\theta _I\}_{I\subset [k]}$ as the restriction data of $\sigma $ .

Lemma 3.46. The projection map $v:\mathcal {L}_X \to X$ is a trivial fibration of MB simplicial sets.

Proof. Since v by definition detects all possible decorations, it will suffice to show that v is a trivial fibration on the underlying simplicial sets. Note that v is a bijection on $0$ -simplices. Given $k \geqslant 1$ we consider a lifting problem

To produce the dotted arrow we use the bottom horizontal morphism as our choice for simplex in X. If $p \circ \hat {\sigma }(k) \neq n$ or $p \circ \hat {\sigma }$ is constant on n, we set $\theta _{[k]}$ to be the unique preimage of $\hat {\sigma }$ in $\mathcal {M}_X$ . If $p \circ \hat {\sigma }(k)=n$ and it is not constant on n, we set $\theta _{[k]}=\varnothing $ . The rest of the $\theta _{I}$ are always chosen according to the top horizontal morphism. The compatibilities are clearly satisfied.

We construct a morphism $u:\mathcal {M}_X \to \mathcal {L}_X$ that sends a simplex $\theta :\Delta ^k \to \mathcal {M}_X$ to the simplex $\omega (\theta )$ in X. For every $I \subseteq [k]$ we set $\theta _I=d_I(\theta )$ . It is clear that u is a cofibration whose image is given by those simplices in $\mathcal {L}_X$ such that $\theta _{[k]}\neq \varnothing $ . It is not hard to see that u induces a bijection on the restriction to $\Delta ^{[0,n-1]}$ and on the fibre over n.

Remark 3.47. Let $\pi =p \circ v: \mathcal {L}_X \to \Delta ^n$ . Given $\sigma :\Delta ^k \to \mathcal {L}_X$ we fix the notation $\overline {\sigma }=\pi \circ \sigma $ .

Definition 3.48. Let $\sigma :\Delta ^k \to \mathcal {L}_X$ be a simplex such that $\overline {\sigma }(k)=n$ . Let $\kappa _{\overline {\sigma }}$ be the first element in $[k]$ such that $\overline {\sigma }(\kappa _{\overline {\sigma }})=n$ . We define a full subposet $Z_{\sigma } \subset [k]\times [n]$ consisting of

• • Those vertices of the form $(x,\overline {\sigma }(x))$ with $x < \kappa _{\overline {\sigma }}$ .

• • Those vertices of the form $(x,y)$ with $x \geqslant \kappa _{\overline {\sigma }}$ and $y \geqslant \overline {\sigma }(0)$ .

We denote by $\mathcal {Z}_\sigma $ the nerve $\operatorname {N}(Z_\sigma )$ . Note that the projection $[k]\times [n]\to [n]$ yields a canonical map $\mathcal {Z}_{\sigma } \to \Delta ^n$ . We endow $\mathcal {Z}_{\sigma }$ with the structure of an $\textbf {MB}$ simplicial set by declaring an edge $(x_1,y_1) \to (x_2,y_2)$ marked if $x_1 =x_2 \geqslant \kappa _{\overline {\sigma }}$ and $y_2=n$ . A triangle is declared to be lean if the associated 2-simplex in $\Delta ^k$ is degenerate. Finally we say that a triangle in $\mathcal {Z}_{\sigma }$ is thin if it is already lean and its image in $\Delta ^n$ is degenerate.

We call those non-degenerate simplices $\rho :\Delta ^\ell \to \mathcal {Z}_{\sigma }$ which are not contained in any other non-degenerate simplex essential. Note that there exists a canonical essential simplex $\Gamma _{\sigma } \colon \Delta ^k \to \mathcal {Z}_\sigma $ given by $x_i \mapsto (x_i,\overline {\sigma }(x_i))$ .

Figure 1 The poset $Z_\sigma $ corresponding to the map $[5]\to [4]$ given by the sequence of values $0,1,1,2,4,4$ .

Remark 3.49. The avid reader might complain that our definition of $\mathcal {Z}_\sigma $ only depends on $\overline {\sigma }$ and so should be denoted by $\mathcal {Z}_{\overline {\sigma }}$ . The next definition will justify our notation.

Definition 3.50. Let $\sigma :\Delta ^k \to \mathcal {L}_X$ such that $\overline {\sigma }(k)=n$ . We define a subsimplicial set $\mathcal {X}_{\sigma }\subset \mathcal {Z}_{\sigma }$ (with the inherited marking and scalings) consisting of those simplices satisfying at least one of the conditions below

1. i) We have $y_i=\overline {\sigma }(x_i)$ for $i=0,\dots ,\ell $ .

2. ii) There exists $I \subseteq [k]$ such that $\theta _I \neq \varnothing $ with $\overline {\sigma }(\max (I))=n$ and $x_i \in I$ for $i=0,\dots ,\ell $ .

Definition 3.51. Let $\sigma :\Delta ^k \to \mathcal {L}_X$ such that $\overline {\sigma }(k)=n$ and suppose we are given a subset $I \subset [k]$ such that $\theta _{I}\neq \varnothing $ and such that $\overline {\sigma }(\max (I))=n$ . We define a subposet $K_I \subset \Delta ^I \times \Delta ^{[\overline {\sigma }(0),n]}$ to be the intersection of $\Delta ^I\times \Delta ^{[\overline {\sigma }(0),n]}$ with $\mathcal {Z}_\sigma $ . We denote $\mathcal {K}_I$ the $\textbf {MB}$ simplicial set obtained by equipping $K_I$ with the decorations induced from $\mathcal {Z}_\sigma $ .

Remark 3.52. Observe that we can construct $\mathcal {X}_\sigma $ as the union of every $\mathcal {K}_I$ inside of $\mathcal {Z}_\sigma $ together with $\Gamma _{\sigma }$ (cf. Definition 3.48).

Construction 3.53. Let $\sigma :\Delta ^k \to \mathcal {L}_X$ such that $\overline {\sigma }(k)=n$ and suppose we are given a subset $I \subset [k]$ such that $\theta _{I}\neq \varnothing $ and such that $\overline {\sigma }(\max (I))=n$ . We construct a morphism

$$\begin{align*}\Delta^I \times \Delta^{[\overline{\sigma}(0),n]}\to (\Delta^n)^{\diamond}\times X_n \end{align*}$$

whose component at $(\Delta ^n)^{\diamond }$ is given by $\Delta ^I \times \Delta ^{[\overline {\sigma }(0),n]} \to \Delta ^{[\overline {\sigma }(0),n]} \to (\Delta ^n)^{\diamond }$ and whose component at $X_n$ is given by $\Delta ^I \times \Delta ^{[\overline {\sigma }(0),n]} \to \Delta ^I \to X_n$ where the last morphism is induced from $\theta _I$ . Note that this map is compatible with the decorations of $\mathcal {Z}_\sigma $ , which implies that we have a morphism $\mathcal {K}_I \to (\Delta ^n)^{\diamond }\times X_n$ induced by restriction.

Remark 3.54. Let $\sigma :\Delta ^k \to \mathcal {L}_X$ such that $\overline {\sigma }(k)=n$ . We define a morphism $\widetilde {f}_\sigma :\mathcal {X}_\sigma \to \mathcal {L}_X$ as follows:

• • For simplices satisfying condition $i)$ in Definition 3.50, $\widetilde {f}_\sigma $ is simply $\sigma $ .

• • For simplices satisfying condition $ii)$ in Definition 3.50, $\widetilde {f}_\sigma $ is given by the composite

  

One observes that this definition is compatible in the various intersections $\mathcal {K}_I \cap \mathcal {K}_J$ thus defining the desired morphism.

Definition 3.55. Let $\sigma :\Delta ^k \to \mathcal {L}_X$ such that $\overline {\sigma }(k)=n$ . We define a subsimplicial set $\mathcal {X}_\sigma ^{\uparrow } \to \mathcal {Z}_\sigma $ (with the induced decorations) consisting of those simplices that are either in $\mathcal {X}_\sigma $ or satisfy the property:

• • There exists $j \in [k]$ such that $x_i\neq j$ for every $i=0,\dots ,\ell $ and such that $\overline {\sigma }(d^j(k-1))=n$ .

Remark 3.56. Note that we can equivalently define $\mathcal {X}_\sigma ^{\uparrow }$ to consist of those simplices that are either in $\mathcal {X}_\sigma $ or that are contained in the image of

$$\begin{align*}\mathcal{Z}_{d_j(\sigma)} \to \mathcal{Z}_\sigma \end{align*}$$

where $d_j(\sigma )(k-1)=n$ .

Definition 3.57. We define an order on the set of essential simplices of $\mathcal {Z}_{\sigma }$ which we denote by ‘ $\prec $ ’. Let $\rho _i$ for $i=1,2$ be two essential simplices determined by the sequence of vertices for $i=1,2$ . Let $\varepsilon $ be the first index such that $\rho _1(\varepsilon )\neq \rho _2(\varepsilon )$ . We say that $\rho _1 \prec \rho _2$ if precisely one of the following conditions is satisfied:

• • We have that $x^1_{\varepsilon }=\kappa _{\sigma }$ .

• • We have $x^i_\varepsilon> \kappa _\sigma $ for $i=1,2$ and $y^1_\varepsilon>y^2_\varepsilon $ .

Figure 2 An essential simplex in $\mathcal {Z}_\sigma $ with $\theta $ and its anterior, recumbent and plumb vertices labelled.

Lemma 3.58. Let $\sigma :\Delta ^k \to \mathcal {L}_X$ such that $\overline {\sigma }(k)=n$ . Then the following morphisms are MB-anodyne:

$$\begin{align*}\mathcal{X}_{\sigma} \to \mathcal{X}_\sigma^{\uparrow} \to \mathcal{Z}_{\sigma}. \end{align*}$$

Proof. If $\sigma $ factors through $\mathcal {M}_X$ , then $\mathcal {X}_\sigma =\mathcal {Z}_\sigma $ , so we may assume without loss of generality that $\sigma $ does not factor through $\mathcal {M}_X$ .

We proceed by induction on k. Consider $k=1$ , and note that $\mathcal {X}_\sigma =\mathcal {X}_\sigma ^\uparrow $ . Since $\sigma $ does not factor through $\mathcal {M}_X$ , we may assume $\sigma (0)\neq n$ . It then follows that the map $\mathcal {X}_\sigma \to \mathcal {Z}_\sigma $ can be identified with the inclusion

$$\begin{align*}\psi_\ell\colon\Delta^1\coprod_{\Delta^0} (\Delta^n)^\diamond \to (\Delta^{n+1})^\dagger, \end{align*}$$

where the pushout identifies the terminal objects in $\Delta ^1$ and $\Delta ^n$ respectively and where $\dagger $ indicates marking where $i\to n+1$ is marked when $i\neq 0$ , where every triangle is lean, and only those over degenerate triangles in $\Delta ^n$ are thin. It follows immediately from Lemma 2.39 that this morphism is MB-anodyne.

We now suppose that the lemma holds in dimension $k-1$ . By the inductive hypothesis,

$$\begin{align*}\mathcal{X}_\sigma\to \mathcal{X}_\sigma^\uparrow \end{align*}$$

is MB-anodyne since this map can be obtained as an iterated pushout of the morphisms $\mathcal {X}_{d_j(\sigma )} \to \mathcal {Z}_{d_j(\sigma )}$ . Thus, it is sufficient for us to show that the morphism $\mathcal {X}_\sigma ^\uparrow \to \mathcal {Z}_\sigma $ is MB-anodyne.

We will add essential simplices of $\mathcal {Z}_\sigma $ according to the order $\prec $ . Given an essential simplex $\rho $ , we denote by $N_\rho $ the MB simplicial subset of $\mathcal {Z}_\sigma $ obtained by adding every essential simplex $\rho ^\prime $ such that $\rho ^\prime \preceq \rho $ . We consider a pullback diagram

and we turn our attention to proving that the top horizontal morphism is MB-anodyne. Let us fix the notation and denote by $\theta $ the first index such that $x_\theta =\kappa _\sigma $ .

We define three types of vertices $\varepsilon \in [r]$ of $\rho $ .

• Anterior vertices are those $\varepsilon $ which have $x_\varepsilon <\kappa _\sigma $ .

• Recumbent vertices are those $\varepsilon \in [r]$ which have $x_\varepsilon>\kappa _\sigma $ and $x_{\varepsilon -1} <x_\varepsilon $ . Note that this necessarily implies $y_{\varepsilon -1}=y_\varepsilon $

• Plumb vertices are those $\varepsilon \in [r]$ which have $x_\varepsilon \geqslant \kappa _\sigma $ and $x_{\varepsilon -1}=x_\varepsilon $ . Note that this necessarily implies that $y_{\varepsilon -1}<y_\varepsilon $ . Note also that every $\rho $ has at least one plumb vertex.

Notice that the only vertex which is not anterior, recumbent or plumb is $\theta $ . We call a vertex $\varepsilon \in [r]$ a downturn if $\varepsilon $ is either recumbent or $\varepsilon =\theta $ and $x_{\varepsilon +1}=x_\varepsilon $ . Note that $\rho $ is uniquely determined by its set of downturns and the fact that it is essential.

We will prove three claims about these types of vertices, which then will enable us to complete the proof.

Claim 1. If $\varepsilon $ is an anterior vertex, then $d_\varepsilon (\Delta ^r)\subset Q_\rho $ .

ProofSubproof. Since $\varepsilon $ is anterior, it is the only vertex of $\rho $ whose first coordinate is $x_\varepsilon $ . Consequently, $d_\varepsilon (\rho )$ factors through $\mathcal {Z}_{d_\varepsilon (\sigma )}$ .

Claim 2. Let $\varepsilon $ be a recumbent vertex. Then $d_\varepsilon (\Delta ^r)\subset Q_\rho $ .

ProofSubproof. There are two cases. If $x_{\varepsilon +1}>x_\varepsilon $ , then as before $d_\varepsilon (\rho )$ factors through $Z_{d_j(\sigma )}$ for some face operator $d_j$ . If, on the other hand, $x_{\varepsilon +1}=x_\varepsilon $ , then $d_\varepsilon (\rho )$ factors through a previous essential simplex.

Claim 3. Let X be a set of plumb vertices. Then $d_X(\Delta ^r)\nsubseteq Q_\rho $ .

ProofSubproof. Since $d_X(\rho )$ contains a point with first coordinate j for every $j\in [k]$ , we see that $d_X(\rho )$ cannot factor through $\mathcal {Z}_{d_j(\sigma )}$ . Similarly, if $d_X(\rho )$ factors through $\mathcal {K}_I$ for $I\subset [k]$ , then $I=[k]$ , which would imply that $\sigma $ factors through $\mathcal {M}_X$ . Moreover, $d_X(\rho )$ cannot factor through $\Gamma _{\sigma }$ , since it contains the vertex $(x_\theta ,y_\theta )$ .

Finally, if $\gamma \prec \rho $ is a previous simplex in our factorisation, then $d_X(\rho )$ cannot factor through $\gamma $ , since $\gamma $ and $\rho $ are uniquely determined by their sequences of downturns, and the only sequence of downturns containing $d_X(\rho )$ determine simplices greater than $\rho $ under the order $\prec $ .

We define $\mathcal {A} \subset \mathbb {P}([r])$ by declaring $\mathcal {A}$ to contain all subsets of the form for every simplex of maximal dimension $\Delta ^J \to Q_\rho $ . Note that the first and second claim above imply that $\{j\} \in \mathcal {A}$ for every anterior or recumbent vertex. Given $U \in \mathcal {A}$ which is not of the form $\{j\}$ as before, then it follows that $\min (U)=\theta $ and that the rest of the elements of U are plumb vertices. We observe that if the successor of $\theta $ is recumbent or if it is of the form $(\kappa _\sigma ,n)$ then $U=\{\theta \}$ . Otherwise the successor of $\theta $ belongs to U. Repeating this argument we see that $U=[\theta ,s]$ where for every $\theta < t \leqslant s$ we have that t is plumb and s is not of the form $(\kappa _\sigma ,n)$ .

To finish the proof we will consider two different cases. Each of these cases will be solved using inner-dull subsets (resp. right-dull) subsets with respect to $\mathcal {A}$ as above. It is important to remark that since we can assume that $\dim (\sigma )> 1$ it follows that all the decorations in $\mathcal {Z}_{\sigma }$ factor through $\mathcal {X}^{\uparrow }_\sigma $ . The first case is given precisely when the vertex r is recumbent. In this situation it follows that we can use Lemma 2.36 where the pivot point is the biggest plumb vertex. Since $r \neq \theta $ it follows that if r is not recumbent it must be plumb. In this case the claim follows from Lemma 2.39.

Figure 3 A depiction of $\mathsf {V}(\sigma )$ .

Definition 3.59. Let $\sigma :\Delta ^k\to \mathcal {L}_X$ such that $\overline {\sigma }(k)=n$ and let $\ell _{\sigma }=n-\overline {\sigma }(\kappa _{\overline {\sigma }}-1)$ . For every morphism $f_\sigma : \mathcal {Z}_{\sigma } \to \mathcal {L}_X$ over $\Delta ^n$ such that its restriction to $\mathcal {X}_\sigma $ equals $\widetilde {f}_\sigma $ as in Remark 3.54, we define a simplex $ \mathsf {B}(\sigma )\colon \Delta ^{k+\ell _{\sigma }}\to \mathcal {L}_X$ to be the composite

where

$$\begin{align*}\mathsf{V}(\sigma) :{[k+\ell_{\sigma}]}\to Z_{\sigma}, \enspace j\mapsto \begin{cases} (j,\overline{\sigma}(j)) & 0\leqslant j\leqslant \kappa_{\overline{\sigma}}-1 \\ (\kappa_{\overline{\sigma}},\overline{\sigma}(\kappa_{\overline{\sigma}}-1)) & j = \kappa_{\overline{\sigma}} \\ (\kappa_{\overline{\sigma}},,\overline{\sigma}(\kappa_{\overline{\sigma}}-1)+j-\kappa_\sigma), & \kappa_{\overline{\sigma}}<j \leqslant \kappa_{\overline{\sigma}}+\ell_{\sigma} \\ (j-\ell_{\sigma},n), & \kappa_{\overline{\sigma}}+\ell_{\sigma}+1 \leqslant j \leqslant k+\ell_{\sigma} \end{cases} \end{align*}$$

Note that the canonical map $\Gamma _\sigma \colon \Delta ^k \to \mathcal {Z}_\sigma $ factors through $\mathsf {V}(\sigma )$ .

Remark 3.60. We can equivalently characterise the simplex $\mathsf {V}(\sigma )$ in Definition 3.59 as the smallest essential simplex of $\mathcal {Z}_\sigma $ under the order $\prec $ of Definition 3.57 that does not factor through $\mathcal {X}_{\sigma }^\uparrow $ .

Remark 3.61. There are two key parameters which we will use to analyse the simplices $\mathsf {B}(\sigma )$ , for $\sigma : \Delta ^k\to \mathcal {L}_X$ . One is the fundamental vertex $\kappa _\sigma $ – the first vertex such that $\overline {\sigma }(\kappa _\sigma )=n$ . The other is the terminal size of $\sigma $ : the number of vertices $j\in [k]$ such that $\overline {\sigma }(j)=n$ . We will denote the terminal size of $\sigma $ by

Notice that the terminal size of $\mathsf {B}(\sigma )$ is always equal to the terminal size of $\sigma $ .

To make use of the simplices $\mathsf {B}(\sigma )$ in an inductive pushout argument, we will need to ensure we can make sufficiently compatible choices of maps

$$\begin{align*}f_\sigma\colon \mathcal{Z}_\sigma \to \mathcal{L}_X \end{align*}$$

to define our choices of $\mathsf {B}(\sigma )$ .

Proposition 3.62. There exists a collection indexed by the non-degenerate simplices of $\mathcal {L}_X$

With the following properties:

1. i) The maps $f_\sigma $ are morphisms over $\Delta ^n$ .

2. ii) The restriction of $f_\sigma $ to $\mathcal {X}_{\sigma }$ equals $\widetilde {f}_\sigma $ as in Remark 3.54.

3. iii) Given a face operator $d^j:[k-1] \to [k]$ such that $\overline {\sigma }(d^j(k-1))=n$ we have that the composite

   

   equals $f_{d_j(\sigma )}$ .

4. iv) Let $\tau =d_S(\sigma )$ where $d_S\colon [s] \to [n]$ is injective. Suppose that we are given a commutative diagram

   

   where $\alpha $ is injective. Then there exists a commutative diagram

   

   where $\omega $ is injective and $s_{\kappa _\sigma -1}$ is a degeneracy operator.

Remark 3.63. Let $\sigma \colon \Delta ^k \to \mathcal {L}_X$ . If we can factor $\sigma =\omega \circ \alpha $ where $\alpha \colon \Delta ^{k} \to \Delta ^{v}$ is injective, we will abusively write $\sigma \subseteq \omega $ .

Remark 3.64. Of key importance to our argument is the fact that, if $\sigma \subseteq \mathsf {B}(\tau )$ and $\tau \subsetneq \sigma $ , then $\overline {\sigma }^{-1}(n)$ , $\overline {\tau }^{-1}(n)$ and $\overline {\mathsf {B}(\tau )}^{-1}(n)$ have the same cardinality. Moreover, further unraveling reveals that $\sigma $ is obtained from $\mathsf {B}(\tau )$ by deleting certain vertices j with the property that $\kappa _{\tau } \leqslant j < \kappa _{\sigma }$ .

Definition 3.65. Set . We will call a collection

a compatible k-collection if it satisfies conditions i)-iv) above.

Lemma 3.66. Let be a compatible $(k-1)$ -collection, and let $\sigma :\Delta ^k\to \mathcal {L}_X$ such that $\sigma $ does not factor through $\mathcal {M}_X$ . Suppose that there is a simplex $\tau :\Delta ^s\to \mathcal {L}_X$ with $s\leqslant k$ such that

• • There is an inclusion $\sigma \subseteq \mathsf {B}(\tau )$ .

• • The terminal sizes agree, i.e. $\nu _\sigma =\nu _{\mathsf {B}(\tau )}$ .

Then there is a subsimplex $\gamma \subseteq \tau $ such that

1. 1. There is an inclusion $\gamma \subseteq \sigma $ which is strict if $s<k$ .

2. 2. There is an inclusion $\sigma \subseteq \mathsf {B}(\gamma )$

3. 3. The terminal sizes $\nu _\sigma $ and $\nu _{\mathsf {B}(\gamma )}$ agree.

Proof. Let $\alpha :\Delta ^{k} \to \Delta ^{s + \ell _\tau }$ witnessing $\sigma \subseteq \mathsf {B}(\tau )$ (see Remark 3.63). We consider a pullback diagram

and set $\sigma \circ \varphi =\gamma $ . Conditions (1) and (3) are immediate, and it is an easy check to see that $\sigma \subseteq \mathsf {B}(\gamma )$ .

Lemma 3.67. Let be a compatible $(k-1)$ -collection. Let $\sigma :\Delta ^k \to \mathcal {L}_X$ and assume that $\sigma $ does not factor through $\mathcal {M}_X$ . Let us suppose there exists a pair $\tau _i:\Delta ^{s_i} \to \mathcal {L}_X$ with $s_i <k$ for $i=1,2$ ; such that $\sigma \subseteq \mathsf {B}(\tau _i)$ and $\tau _i \subsetneq \sigma $ . Then $\tau _1 \subseteq \tau _2$ or $\tau _2 \subseteq \tau _1$ .

Proof. We can partition $\Delta ^{s_i}$ into two parts: $\tau _i^{-1}([0,n-1])$ and $\tau _i^{-1}(n)$ . We identify each of these with subsets of $\Delta ^k$ . By Remark 3.64, $\tau _1^{-1}(n)=\tau _2^{-1}(n)=\sigma ^{-1}(n)$ . Since $\sigma $ does not factor through $\mathcal {M}_X$ , the initial vertex of $\sigma $ must factor through each $\tau _i$ . Since the only vertices j of $\mathsf {B}(\tau _i)$ which are not vertices of $\tau _i$ satisfy $j\geqslant \kappa _{\tau _i}$ , we see that for $j\in [k]$ such that $j<\kappa _{\tau _i}$ , $\sigma (j)$ must factor through $\tau _i$ .

Since $\tau _i$ is obtained from $\sigma $ by deleting the vertices $j\in [k]$ such that $\kappa _{\tau _i}\leqslant j<\kappa _\sigma $ (cf. Remark 3.64) we see that if $\kappa _{\tau _1}\leqslant \kappa _{\tau _2}$ , then $\tau _1\subseteq \tau _2$ .

Corollary 3.68. Let be a compatible $(k-1)$ -collection, and let $\sigma \colon \Delta ^k\to \mathcal {L}_X$ be a simplex that does not factor through $\mathcal {M}_X$ . We define $\Omega _{\sigma }$ to be the poset consisting in those simplices $\tau \colon \Delta ^s\to \mathcal {L}_X$ with $\sigma \subseteq \mathsf {B}(\tau )$ and $\tau \subsetneq \sigma $ , ordered by inclusion (cf Remark 3.63). If $\Omega _{\sigma }\neq \varnothing $ then $\Omega _{\sigma }$ admits a minimal element $\tau _{\min }$ . Moreover, given $\tau \in \Omega _{\sigma }$ such that $\sigma =\mathsf {B}(\tau )$ , then $\tau =\tau _{\min }$ .

Proof. The first statement is an immediate consequence of the previous lemma. To prove the second claim suppose that we have $\rho \subseteq \tau $ such that $\mathsf {B}(\tau ) \subseteq \mathsf {B}(\rho )$ . First we observe that $\nu _\tau =\nu _\rho $ . Let $\varepsilon _{\tau }=\overline {\tau }(\kappa _{\tau }-1)$ and define the parameter $\ell _{\tau }$ as the number of vertices in $\mathsf {B}(\tau )$ living over $\varepsilon _{\tau }$ . We also denote by $\ell _{\rho }$ the number of vertices in $\mathsf {B}(\rho )$ living over $\varepsilon _{\tau }$ and observe that since $\rho \subseteq \tau $ then we must have $\ell _{\rho } \leqslant \ell _{\tau }$ . Moreover, since $\mathsf {B}(\tau ) \subseteq \mathsf {B}(\rho )$ it follows that $\ell _{\rho }=\ell _{\tau }$ . This implies that $\kappa _{\tau }-1 \in \rho $ . We can now easily verify that if $\rho $ is obtained from $\tau $ by deleting some vertices which are strictly smaller than $\kappa _{\tau }-1$ then it follows that the dimension $\mathsf {B}(\rho )$ is strictly smaller than that of $\mathsf {B}(\tau )$ hence $\rho =\tau $ .

Definition 3.69. Suppose we are given a compatible $(k-1)$ -collection and $\sigma :\Delta ^k\to \mathcal {L}_X$ . If it exists, we call the minimal simplex of Corollary 3.68 the capsule of $\sigma $ . We say that $\sigma $ is encapsulated if it admits a capsule.

There is one final fact to establish: that there is a way of choosing a compatible degeneracy to ensure condition iv). Given $\sigma : \Delta ^k\to \mathcal {L}_X$ which does not factor through $\mathcal {M}_X$ , we denote by $\mathcal {R}_\sigma $ the pullback

Given a compatible $(k-1)$ -collection , the compatibilities (1) and (2) allow us to define a map

$$\begin{align*}\tilde{f}_\sigma^\uparrow\colon \mathcal{X}_\sigma^\uparrow \to \mathcal{L}_X \end{align*}$$

for each $\sigma :\Delta ^k\to \mathcal {L}_X$ which extends $\tilde {f}_\sigma $ , and which agrees with $f_{d_j(\sigma )}$ for each face operator $d_j$ such that $d_j(\sigma )(k-1)=n$ .

Lemma 3.70. Let be a compatible $(k-1)$ -collection with $k\geqslant 3$ , and suppose that $\sigma :\Delta ^k\to \mathcal {L}_X$ is encapsulated with capsule $\tau \colon \Delta ^{s} \to \mathcal {L}_X$ . Then for each $\zeta :\Delta ^r\to \mathcal {R}_\sigma $ such that $\zeta $ hits both $\mathsf {V}(\sigma )(\kappa _\sigma -1)$ and $\mathsf {V}(\sigma )(\kappa _\sigma )$ , then $\widetilde {f}_\sigma ^\uparrow \circ \zeta $ is degenerate on those vertices.

Proof. Note that our assumption means that $\zeta $ does not factor through $\sigma $ .

First suppose that $\zeta $ factors through $\mathcal {Z}_{d_j(\sigma )}$ for $j\leqslant \kappa _\tau -1$ . Then we note that $d_j(\sigma )\subset \mathsf {B}(d_j(\tau ))$ , and $\nu _{d_j(\sigma )}=\nu _{\mathsf {B}(d_j(\tau ))}$ , so by Lemma 3.66 and the fact that satisfies iv) we see that $\tilde {f}_\sigma ^\uparrow \circ \zeta $ is degenerate on the desired vertices. An identical argument holds when $j>\kappa _\sigma $ .

If $\zeta $ factors through $\mathcal {Z}_{d_j(\sigma )}$ for $\kappa _\tau \leqslant j\leqslant \kappa _\sigma $ , then $\sigma (j)$ is not in $\tau $ . Thus $\tau \subset d_j(\sigma )$ , $d_j(\sigma )\subset \mathsf {B}(\tau )$ , and so since satisfies condition iv), $\zeta $ degenerates accordingly.

Finally, suppose that $\zeta $ factors through for some $\theta _I^\sigma \neq \varnothing $ . Then $I\cap [s]=J$ has $\theta ^\tau _J\neq \varnothing $ , and we can factor $\tilde {f}_\sigma ^\uparrow \circ \zeta $ through u as

$$\begin{align*}\Delta^r\to \Delta^I\times \Delta^{[\sigma(0),n]}\to \Delta^n\times X_n \end{align*}$$

By construction, the first factor of this simplex is degenerate at the desired vertex. The second factor can be equivalently factored through $\Delta ^J\times \Delta ^{[\tau (0),n]}$ and thus is degenerate at the desired vertex as well. Thus $\tilde {f}_\sigma ^\uparrow \circ \zeta $ is degenerate.

Corollary 3.71. Let be a compatible $(k-1)$ -collection with $k \geqslant 3$ . Suppose that $\sigma :\Delta ^k\to \mathcal {L}_X$ is encapsulated, and let $\tau $ be the capsule for $\sigma $ . We define $\gamma \subset \mathsf {B}(\tau )$ by deleting the vertices $j \in \mathsf {B}(\tau )$ such that such that $j\leqslant \kappa _{\sigma }-1$ and $j \notin \sigma $ . Then the diagram

commutes.

With this corollary in hand we can now return to Proposition 3.62.

Proof of Proposition 3.62.

First let us observe that the choices of $f_\sigma $ for every $\sigma : \Delta ^k \to \mathcal {M}_X \subset \mathcal {L}_X$ are already made since $\mathcal {X}_\sigma = \mathcal {Z}_\sigma $ . It is also easy to check that the rest of the conditions hold for those choices. Therefore we can restrict our attention to producing the choices for simplices $\sigma :\Delta ^k \to \mathcal {L}_X$ that do not factor through $\mathcal {M}_X$ .

We will inductively define compatible k-collections for every $k\geqslant 1$ . Before commencing our argument we will make a preliminary definition. Given $\sigma :\Delta ^{k} \to \mathcal {L}_X$ we define $\mathcal {Y}^{\uparrow }_\sigma $ to be the simplicial subset (with the inherited decorations) of $\mathcal {Z}_\sigma $ whose simplices are those of $\mathcal {X}^{\uparrow }_\sigma $ in addition to the simplex $\mathsf {V}(\sigma )$ . It follows from the argument given in Lemma 3.58 that the inclusion $\mathcal {Y}^{\uparrow }_\sigma \to \mathcal {Z}_\sigma $ is MB-anodyne.

For every $e:\Delta ^1 \to \mathcal {L}_X$ we fix the choice of $f_e$ which is guaranteed by Lemma 3.58. In this ground case, there are no conditions to check. Let us consider a triangle $\sigma :\Delta ^2 \to \mathcal {L}_X$ . Using the previous choices the can extend the map $\widetilde {f}_\sigma $ to a morphism

$$\begin{align*}f^\uparrow_\sigma\colon \mathcal{X}_\sigma^{\uparrow} \to \mathcal{L}_X \end{align*}$$

We distinguish now two cases. Suppose that $\sigma $ is not contained in some $\mathsf {B}(e)$ for $e:\Delta ^1 \to \mathcal {L}_X$ . Then we define $f_\sigma $ to be an extension of $f^{\uparrow }_\sigma $ to $\mathcal {Z}_\sigma $ . If $\sigma \subseteq \mathsf {B}(e)$ we can assume that $e \subset \sigma $ since otherwise we have $\sigma \in \mathcal {M}_X$ (or alternatively invoking Lemma 3.66). We extend $f^{\uparrow }_\sigma $ to a map $\mathcal {Y}^\uparrow _\sigma \to \mathcal {L}_X$ by sending $\mathsf {V}(\sigma )$ to the following simplex: Let $\sigma _e: \Delta ^r \to \mathcal {L}_X$ be the simplex obtained by forgetting every vertex j in $\mathsf {B}(e)$ such that $j\leqslant \kappa _{\overline {\sigma }}-1$ and such that is not in $\sigma $ . We can now set $\mathsf {B}(\sigma )$ to be given by $s_\alpha (\sigma _e)$ where $\alpha =\kappa _{\overline {\sigma }}-1$ and consequently condition $iv)$ is satisfied. This means that we can construct a compatible 2-collection .

Now suppose we have a compatible $(k-1)$ -collection . Let $\sigma : \Delta ^k\to \mathcal {L}_X$ be a simplex. If $\sigma $ is not encapsulated, then we may define $f_\sigma $ by solving the lifting problem

using 3.58. If $\sigma $ is encapsulated with capsule $\tau $ , we can use Corollary 3.71 to define a map

$$\begin{align*}\mathcal{Y}_\sigma^\uparrow \to \mathcal{L}_X \end{align*}$$

which sends $\mathsf {B}(\sigma )$ to the degenerate simplex described in Corollary 3.71. Solving the corresponding lifting problem yields an $f_\sigma $ satisfying i)-iv). Thus, we can extend to a compatible k-collection, as desired.

Proposition 3.72. The cofibration $u:\mathcal {M}_X \to \mathcal {L}_X$ is MB-anodyne.

Proof. We say that a simplex $\sigma :\Delta ^k \to \mathcal {L}_X$ is wide if it is not contained in the image of u. Let $\sigma : \Delta ^k \to \mathcal {L}_X$ and recall the definition . We produce a filtration

$$\begin{align*}\mathcal{M}_X \to S^1 \to S^2 \to \cdots \to \mathcal{L}_X \end{align*}$$

where $S^\alpha $ consists of those simplices $\sigma $ in $\mathcal {L}_X$ that either factor through $\mathcal {M}_X$ or satisfy $\nu _\sigma \leqslant \alpha $ . We will fix the convention $S^0=\mathcal {M}_X$ . We will show that each step in the filtration is MB-anodyne. Let us fix once and for all a choice of $f_\sigma :\mathcal {Z}_\sigma \to \mathcal {L}_X$ for every $\sigma :\Delta ^k \to \mathcal {L}_X$ with the properties listed in Proposition 3.62. First, let us observe that given $\sigma :\Delta ^k \to S^\alpha $ it follows that the morphisms $f_\sigma $ also factor through $S^\alpha $ . We can now define $S^{(\alpha ,s)}$ to consist of those simplices contained in $S^{\alpha }$ in addition to the simplices $B(\sigma )$ for $\sigma :\Delta ^k \to \mathcal {L}_X$ wide and non-degenerate, such that $k \leqslant s$ and $\nu _\sigma =\alpha +1$ . This produces a filtration

$$\begin{align*}S^{\alpha-1} \to S^{(\alpha-1,\alpha)} \to S^{(\alpha-1,\alpha+1)} \to \cdots \to S^\alpha. \end{align*}$$

We pick an ordering for the set of non-degenerate simplices of $S^{(\alpha ,s)}$ which do not factor through $S^{(\alpha ,s-1)}$ thus yielding a filtration

$$\begin{align*}S^{(\alpha,s-1)} \to W_{0}\to W_1 \to \cdots \to S^{(\alpha,s)}. \end{align*}$$

Let us consider a pullback diagram

and observe that $A_{\sigma _i}$ already contains the following $(s+\ell _{\sigma _i}-1)$ -dimensional faces:

• • The face that misses the vertex j for $0\leqslant j \leqslant \kappa _{\sigma _i}-1$ . This is because this simplex either factors through $\mathcal {Z}_{d_j(\sigma )}$ if $\kappa _{\sigma _i}\neq 1$ , or it is contained in $\mathcal {M}_X$ otherwise.

• • The face that misses the vertex j for $\kappa _{\sigma _i}+\ell _{\sigma _{i}} \leqslant j \leqslant s +\ell _{\sigma _{i}}$ . This is because those faces have strictly smaller parameter $\nu _{d_j(\sigma _{i})}$ if $\nu _{\sigma _{i}}>1$ or they are already in $\mathcal {M}_X$ if $\nu _{\sigma _{i}}=1$ .

Suppose that there exists a simplex of maximal dimension $\theta \colon \Delta ^{t} \to A_{\sigma _i}$ which is not of the form described before. Then it follows that $\sigma _i \subseteq \theta $ and that $\theta \subseteq \mathsf {B}(\psi )$ for some $\psi \colon \Delta ^{q} \to \mathcal {L}_X$ where $q < s$ or $q=s$ and $\psi < \sigma _i$ . Invoking Lemma 3.66 we see that if $q<s$ then $\sigma \in S^{(\alpha ,s-1)}$ . If $q=s$ then $\sigma _i=\psi $ which contradicts the fact that $\psi < \sigma _i$ . It follows that we can apply Lemma 2.36 where $\mathcal {A}$ consists of the subsets such that $\Delta ^{I} \to A_{\sigma }$ is a simplex of maximal dimension and where the pivot point is given by $\kappa _{\overline {\sigma }}$ . We conclude that the top horizontal morphism in our pullback diagram is MB-anodyne.

We can distill the key upshot of the preceding technical arguments into a single, simple corollary.

Corollary 3.73. For any 2-Cartesian fibration $X\to \Delta ^n_\flat $ , the square

is a homotopy pushout.

Remark 3.74. Observe that in Proposition 3.72 we only use anodyne maps of the form

$$\begin{align*}\bigl(\Lambda^n_i,\{\Delta^{\{i-1,i,i+1\}}\}\bigr)\rightarrow \bigl(\Delta^n,\{\Delta^{\{i-1,i,i+1\}}\}\bigr), \quad n \geqslant 2, \quad 0 < i < n. \end{align*}$$

Consequently, we see that the pushout in Corollary 3.73 is also a pushout in the model structure ${\operatorname {Set}}_{\Delta }^{\mathbf {sc}}$ (see Theorem 2.14).

Remark 3.75. Let $(\Delta ^n,T)=\Delta ^n_T$ be a scaled simplicial set and let $p \colon X \to \Delta ^n_T$ be a 2-Cartesian fibration. Then it follows that we have a homotopy pushout square of scaled simplicial sets

where $T_n$ is the scaling induced by the map $d_n\colon [n-1] \to [n]$ . To verify the claim we denote the strict pushout by $(\mathcal {M}_{X})_{T}$ and consider another pushout diagram

where the top horizontal (and hence the bottom) map is a weak equivalence due to our previous discussion. It is routine to verify that the map $P \to X$ is in the weakly saturated class of morphisms of type (ii) in Definition 2.10.

Proposition 3.76. Let $f \colon (\Lambda ^n_i)_S= (\Lambda ^n_i,S) \to (\Delta ^n,T)=\Delta ^n_T$ be a map of scaled simplicial sets with $i \neq n$ . Given a 2-Cartesian fibration $p \colon X \to \Delta ^n_T$ , then we have a homotopy pushout diagram

where $\Delta ^{n-1}_{S} \subset (\Lambda ^n_i,S)$ is the face that skips the vertex n.

Proof. Note that the map from the pushout of the diagram to $X\times_{\Delta^n_T}(\Lambda^n_i,S)$ remains a weak equivalence after restriction to each of the faces comprising $ (\Lambda ^n_i)_S$ (see Remark 3.75). Since both the pushout and $X\times_{\Delta^n_T}(\Lambda^n_i,S)$ are homotopy pushouts of their restrictions to the faces of $\Lambda ^n_i$ the result follows from the universal property of the homotopy pushout.

3.5.1. The equivalence over a simplex

Having now established the necessary preliminaries, we turn to the proof that the straightening is an equivalence over the minimally-scaled simplex. With few exceptions, the arguments from here on out are standard, and follow the general shape of the analogous arguments given in [23] and [25]. We begin with a lemma, which allows us to more easily apply the straightening to our homotopy pushout.

Lemma 3.77. Consider the inclusion $(\Delta ^{n-1})^\diamond \to (\Delta ^n)^\diamond $ as a morphism in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{\Delta ^n_\flat }$ . Then for every $0\leqslant i<n$ , the induced morphism

$$\begin{align*}\psi \colon \mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}((\Delta^{n-1})^\diamond)(i) \to \mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}((\Delta^{n})^\diamond)(i) \end{align*}$$

is an equivalence of marked-scaled simplicial sets.

Proof. To begin, we examine the morphism on underlying marked simplicial sets. Consider the pushouts

and

and the induced map

$$\begin{align*}\phi\colon{\mathfrak{C}}^{{\operatorname{sc}}}[X](i,\ast) \to {\mathfrak{C}}^{{\operatorname{sc}}}[Y](i,\ast) \end{align*}$$

We first note that, since $i<n$ , we have that ${\mathfrak {C}}^{{\operatorname {sc}}}[X](i,\ast )=\mathfrak {C}[((\Delta ^{n-1})^\diamond )^{\triangleright }](i,\ast )$ .

From the definition, we then have that

$$\begin{align*}{\mathfrak{C}}^{{\operatorname{sc}}}[X](i,\ast)\cong \operatorname{N}(\mathbb{P}(\{i+1,\ldots,n-1\}))^\flat \end{align*}$$

and

$$\begin{align*}{\mathfrak{C}}^{{\operatorname{sc}}}[Y](i,\ast)\cong \operatorname{N}(\mathbb{P}(\{i+1,\ldots,n\}))^\dagger \end{align*}$$

where $\dagger $ indicates the marking in which precisely the non-degenerate morphisms $S\to S\cup \{n\}$ are marked.

We note that, on underlying marked simplicial sets, this means that $\phi $ can be identified with the morphism

$$\begin{align*}{\mathfrak{C}}^{{\operatorname{sc}}}[X](i,\ast)\times \{0\} \to {\mathfrak{C}}^{{\operatorname{sc}}}[X](i,\ast)\times (\Delta^1)^\sharp. \end{align*}$$

We will show that this yields an equivalence of marked-scaled simplicial sets by showing that both scalings are equivalent to the maximal scaling.

We claim that the morphisms

$$\begin{align*}f_n^i\colon\mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}((\Delta^n)^\diamond)(i) \to ( \mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}((\Delta^n)^\diamond)(i))_\sharp \end{align*}$$

and

$$\begin{align*}g_n^i \colon\mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}((\Delta^{n-1})^\diamond)(i) \to ( \mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}((\Delta^{n-1})^\diamond)(i))_\sharp \end{align*}$$

are MS-anodyne. To show that $f_n^i$ is MS-anodyne it suffices to apply Lemma 3.12. The argument for $g_n^i$ is similar and left as an exercise. We thus obtain, for any $i<n$ a commutative diagram

showing that $\phi $ is an equivalence of marked-scaled simplicial sets by 2-out-of-3.

Lemma 3.78. Let $X\to \Delta ^n_\flat $ be a 2-Cartesian fibration, and denote by $X_i$ the fibre over i. Let $\mathbb {S}\!{\operatorname {t}}_*$ denote the straightening over $\Delta ^0$ . Then the map

$$\begin{align*}\psi_i^X\colon\mathbb{S}\!{\operatorname{t}}_*(X_i)\to \mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}(X)(i) \end{align*}$$

is an equivalence of marked-scaled simplicial sets.

Proof. Following [23, 3.2.3.3], we proceed by induction on n. We have already shown the case $n=0$ in Corollary 3.41

By construction, $\psi ^X_n$ is an isomorphism. For $i< n$ , we get a canonical commutative diagram

We can identify the upper-left map with $\psi _i^{X|_{\Delta ^{n-1}}}$ , and so by the inductive hypothesis, it is an equivalence. It thus suffices for us to show that $\gamma _i$ is an equivalence.

By Corollary 3.73, we get a homotopy pushout diagram

in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/\Delta ^n_\flat }$ . Applying the left Quillen functor $\mathbb {S}\!{\operatorname {t}}_{\Delta ^n_\flat }$ yields a homotopy pushout diagram

We have a commutative diagram

where the vertical maps are equivalences of marked-scaled simplicial sets by Theorem 3.15. It thus suffices to note that, by Lemma 3.77, the induced morphism

$$\begin{align*}\psi_i\colon\mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}((\Delta^{n-1})^\diamond)(i) \to \mathbb{S}\!{\operatorname{t}}_{\Delta^n_\flat}((\Delta^{n})^\diamond)(i) \end{align*}$$

is an equivalence for any $i<n$ .

Before continuing, we fix some notation to ease the coming discussion. We will in the following theorem denote the straightening-unstraightening equivalence over the point by

Proposition 3.79. The Quillen adjunction

is a Quillen equivalence.

Proof. As in [23, Lem. 3.2.3.2], we see that $\mathbb {U}\!{\operatorname {n}}_{\Delta ^n_\flat }$ reflects weak equivalences between the images of fibrant objects. It is thus sufficient to show that the derived adjunction unit

$$\begin{align*}\operatorname{Id}\Rightarrow \mathsf{R}(\mathbb{U}\!{\operatorname{n}}_{\Delta^n_\flat})\circ \mathbb{S}\!{\operatorname{t}}_{\Delta^{n}_\flat} \end{align*}$$

is an equivalence. Since $\mathsf {R}(\mathbb {U}\!{\operatorname {n}}_{\Delta ^n_\flat })$ preserves weak equivalences and $\mathbb {S}\!{\operatorname {t}}_{\Delta ^{n}_\flat }$ preserves trivial cofibrations, it is sufficient to check this for fibrant objects.

Let $X\to \Delta ^n_\flat $ be a 2-Cartesian fibration, and let

be a fibrant replacement in $({\operatorname {Set}}_\Delta ^{\mathbf {ms}})^{{\mathfrak {C}}^{{\operatorname {sc}}}[\Delta ^n_\flat ]^{\operatorname {op}}}$ . We are thus left to show that the induced map

is an equivalence in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/\Delta ^n_\flat }$ . Since both objects are fibrant, it suffices to show by Corollary 2.29, that this map is a fibrewise equivalence.

We can identify with . Using the equivalence of Proposition 3.41, we see that the map

is an equivalence if and only if the adjoint map

is an equivalence. However, we can factor this map as

The upper-right map is an equivalence since was a fibrant replacement, and $\psi _i^X$ is an equivalence by Lemma 3.78. The proposition is thus proven.

Corollary 3.80. Consider the scaled simplicial set $(\Delta ^2)_\sharp :={\operatorname {N}}^{\operatorname {sc}}([2])$ . Then Quillen adjunction

is a Quillen equivalence.

Proof. The key point to note is that base change along the cofibration

$$\begin{align*}(\Delta^2)^\sharp_{\flat\subset \sharp}\to (\Delta^2)^\sharp_{\sharp\subset \sharp} \end{align*}$$

induces a fully faithful inclusion

$$\begin{align*}({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/\Delta^2_\flat}^\circ \to ({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/\Delta^2_\sharp}^\circ \end{align*}$$

and similarly, composition with the induced map ${\mathfrak {C}}^{{\operatorname {sc}}}[\Delta ^2_\flat ]\to {\mathfrak {C}}^{\operatorname {sc}}[\Delta ^2_\sharp ]$ induces a fully faithful inclusion

$$\begin{align*}\left(({\operatorname{Set}}_\Delta^{\mathbf{ms}})^{{\mathfrak{C}}^{\operatorname{sc}}[\Delta^2_\sharp]^{\operatorname{op}}}\right)^\circ \to \left(({\operatorname{Set}}_\Delta^{\mathbf{ms}})^{{\mathfrak{C}}^{\operatorname{sc}}[\Delta^2_\flat]^{\operatorname{op}}}\right)^\circ \end{align*}$$

and so we obtain a commutative diagram

of simplicial categories.

The remainder of the proof is, mutatis mutandis, that of [25, Prop. 3.8.7].

3.6. Straightening in general

We now prove the main theorem of this paper.

Theorem 3.81. Let $S\in {{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ be a scaled simplicial set, and let $\phi :{\mathfrak {C}}^{{\operatorname {sc}}}[S]\to \mathcal {C}$ be an equivalence of ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories. The Quillen adjunction

is a Quillen equivalence.

Coupled with the fact, discussed immediately hereafter, that $\mathbb {U}\!{\operatorname {n}}_\phi $ is a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor, this will immediately imply a stronger result – the functor of ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories of fibrant-cofibrant objects induces an equivalence of $\infty $ -bicategories.

The argument from here on out is standard, and follows the same path as [25, Section 3.8]. Our first aim will be to show that, for any scaled simplicial set S, the functor

$$\begin{align*}\mathbb{U}\!{\operatorname{n}}_{\phi}\colon ({\operatorname{Set}}_\Delta^{\mathbf{ms}})^{{\mathfrak{C}}^{{\operatorname{sc}}}[S]^{\operatorname{op}}} \to ({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/S} \end{align*}$$

is, in fact, a ${\operatorname {Set}}_\Delta ^+$ -enriched functor.

The ${\operatorname {Set}}_\Delta ^+$ -enrichment on $\mathbb {U}\!{\operatorname {n}}_{\phi }$ is given as follows. Let be ${\operatorname {Set}}_\Delta ^+$ -enriched functors, and $K\in {{\operatorname {Set}}_{\Delta }^+}$ . A map

is equivalently a map , where . We then have a natural map

Where the second component is induced by the natural transformation $\alpha :\mathbb {S}\!{\operatorname {t}}_*\Rightarrow L$ . We can then write down a natural composite map

Which is equivalently a map . The naturality guarantees that this defines a map of simplicial sets

Similarly, since the composition maps in both cases are defined via the diagonal $\Delta ^n\to \Delta ^n\times \Delta ^n$ , naturality ensures that this defines an enriched functor. A wholly analogous argument shows that $\mathbb {U}\!{\operatorname {n}}_S$ can also be viewed as a simplicially-enriched functor.

Proof of Theorem 3.81.

The proof is now nearly identical to that of [25, Prop. 3.8.4]. The argument hangs on the claim that the functor

sends pushouts along cofibrations to homotopy pullbacks, and sends transfinite composites of cofibrations to homotopy limits, which follows from the argument given in loc. cit.

Corollary 3.82. Let $S\in {{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ be an $\infty $ -bicategory. The ${\operatorname {Set}}_\Delta ^+$ -enriched functor $\mathbb {U}\!{\operatorname {n}}_S$ induces an equivalence of $\infty $ -bicategories

$$\begin{align*}{\operatorname{N}}^{\operatorname{sc}}\left(\left(({\operatorname{Set}}_\Delta^{\mathbf{ms}})^{{\mathfrak{C}}^{{\operatorname{sc}}}[S]^{\operatorname{op}}}\right)^\circ\right) \to {\operatorname{N}}^{\operatorname{sc}}\left(\left(({\operatorname{Set}}_\Delta^{\mathbf{mb}})_{/S}\right)^\circ\right). \end{align*}$$

Proof. This follows immediately from Theorem 3.15, Theorem 3.81, and [23, A.3.1.10].

One final step is left: to interpret this result internally to marked-scaled simplicial sets.

Definition 3.83. The ${\operatorname {Set}}_\Delta ^+$ -enrichment on ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ equips the full subcategory $({{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }})^\circ $ of fibrant-cofibrant objects with the structure of a fibrant ${{\operatorname {Set}}_{\Delta }^+}$ -enriched category. We denote by the homotopy-coherent scaled nerve of this ${{\operatorname {Set}}_{\Delta }^+}$ -category (considered as a scaled simplicial set). We refer to as the $\infty $ -bicategory of $\infty $ -bicategories.

Similarly, for $S\in {{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ , we denote by the $\infty $ -bicategory of 2-Cartesian fibrations over S.

Remark 3.84. Formally, considering ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ as the category of all -small marked-scaled simplicial sets for some Grothendieck universe , the marked-scaled simplicial set is no longer small. We thus resort to fixing a new Grothendieck universe in which , and thus , becomes -small.

Proposition 3.85. Let be a small ${{\operatorname {Set}}_{\Delta }^+}$ -enriched category, S a small scaled simplicial set, an equivalence of ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories, and $\mathbf {A}$ a combinatorial, ${{\operatorname {Set}}_{\Delta }^+}$ -enriched model category. Endow with the projective model structure. Then the functor

is a bicategorical equivalence of scaled simplicial sets.

Proof. The proof is that of [23, Prop. 4.2.4.4]. The only thing that changes is the exchange of ${\operatorname {Set}}_\Delta $ for ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ , and as both of these are excellent model categories, no further emendation is necessary.

Corollary 3.86. Let $S\in {{\operatorname {Set}}_{\Delta }^{\mathbf {sc}}}$ . There is an equivalence of $\infty $ -bicategories

Many of the corollaries of the $(\infty ,1)$ -categorical straightening-unstraightening equivalence generalise directly. In particular, the following results generalise Corollary 3.3.1.1, Corollary 3.3.1.2 and Proposition 3.3.1.7 from [23].

Corollary 3.87. Let $f:T\to S$ be a bicategorical equivalence between scaled simplicial sets. Then the pullback functor

$$\begin{align*}f^\ast\colon ({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/S} \to ({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/T} \end{align*}$$

is a right Quillen equivalence.

Proof. In this case, three of the four functors in the commutative square of Remark 3.9 induce equivalences on underlying $\infty $ -categories. As a result, the right derived functor of $f^\ast $ is an equivalence.

Corollary 3.88. Let $p:X\to S$ be a 2-Cartesian fibration between scaled simplicial sets, and $f:S\to T$ a bicategorical equivalence of scaled simplicial sets. Then there is a 2-Cartesian fibration $q: Y\to T$ and an equivalence of 2-Cartesian fibrations over S, $X\simeq S\times _T Y$ .

Corollary 3.89. Let S be a scaled simplicial set, and let $p:X\to S$ be a fibrant object in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/S}$ . Then the underlying map of scaled simplicial sets is a bicategorical fibration.

Proof. The proof is nearly verbatim that of [23, Prop. 3.3.1.7].

We close the section by providing another useful application of our main theorem.

Theorem 3.90. Let be a 2-Cartesian fibration between $\infty $ -bicategories. Then the pullback functor

is left Quillen where $\left ({\operatorname {Set}}^{\mathbf {sc}}_{\Delta }\right )_{/S}$ is equipped with the model structure where a morphism over X is a weak equivalence (resp. cofibration, resp. fibration) if and only if its image in ${\operatorname {Set}}_\Delta ^{\mathbf {sc}}$ under the forgetful functor is a weak equivalence (resp. cofibration, resp. fibration) in the model structure given in Theorem 2.14.

Proof. It is easy to verify by direct inspection that $p^*$ preserves cofibrations and colimits. Let $f \colon A \to B$ be a weak equivalence in and consider the pullback square

If both $A,B$ are $\infty $ -bicategories then it follows from Corollary 3.89 that the diagram above is a homotopy pullback and the claim holds. Therefore, we can assume without loss of generality that $f \colon A \to B$ is one of the generators in Definition 2.10.

Let $\hat {B}$ be the restriction of B to $\Delta ^{[0,n-1]}$ (resp. $\Delta ^{[0,n-1]}\coprod _{\Delta ^{[0,1]}}\Delta ^0$ ) equipped with decorations induced from B. We claim that we have homotopy pushout squares

where we are denoting by the fibre of over the vertex n. Note that given this claim the result follows immediately from the universal property of the homotopy pushout. We proceed now case by case.

If f is of type (i)-(ii) in Definition 2.10 the claim follows from Remark 3.74, Remark 3.75 and Proposition 3.76. To finish the proof we will assume that f is of type (iii). Let $\hat {f} \colon (\Lambda ^n_0)_{T}=(\Lambda ^n_0,\Delta ^{\{0,1,n\}}) \to (\Delta ^n,\Delta ^{\{0,1,n\}})=\Delta ^n_T$ . Then our previous argument shows that we have homotopy pushout squares

where we are using the composite map

in order to define the corresponding pullbacks above. Finally, we consider a commutative cube

where we observe that the the left-most and right-most faces are homotopy pushouts in addition to the top face. We conclude that the bottom face is also a homotopy pushout. We can use the exact same argument for the remaining pushout square in our claim. The result now follows.

4. The relative 2-nerve

There is a special case of most $\infty $ -categorical Grothendieck constructions in which the computation of the right adjoints can be greatly simplified. When the base is suitably strict, it is possible to define a relative nerve, which computes the Grothendieck construction of a functor. The aim of this appendix is to provide a relative nerve construction which takes as input a ${\operatorname {Set}}_\Delta ^+$ -enriched functor

$$\begin{align*}F\colon{\mathbb{C}}^{{\operatorname{op}}}\to {\operatorname{Set}}_\Delta^{\mathbf{ms} } \end{align*}$$

and yields as output a 2-Cartesian fibration . In form, this relative nerve will actually seem slightly more complicated than the associated straightening functor. However, it will enable us to more easily make the comparison with the strict 2-categorical relative nerve construction of [8]. The particular virtue of our relative 2-nerve construction in this regard is that, given a strict 2-functor

$$\begin{align*}F\colon{\mathbb{C}}^{\operatorname{op}}\to 2\!\operatorname{Cat}, \end{align*}$$

we can compute the relative 2-nerve in terms of strict 2-functors into $\mathbb {C}$ and $F(x)$ , without first passing to simplicial sets.

In our previous papers [4] and [3], we defined two variants of the relative 2-nerve, which provided $\infty $ -bicategories fibred in $(\infty ,1)$ -categories. In this section, we will upgrade the latter of these constructions to provide the desired .

Remark 4.1. Our choice of notation for the relative 2-nerve of a functor $F:{\mathbb {C}}^{{\operatorname {op}}} \to {\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ does in fact collide with the choice of notation in [4] and [3]. An ideal choice of notation would involved a superscript where $\varepsilon $ denotes one of the four variances for bicategorical fibrations. We will use this rather abusive notation to improve readability since we will only consider the outer Cartesian variance.

Definition 4.2. Given a totally ordered set I, the 2-category has

• • Objects given by subsets $S \subseteq I$ such that $\min (S)=i$ .

• • Each mapping category is a poset whose objects $\mathcal {U}\colon S \to T$ are given by subsets $\mathcal {U} \subseteq I$ such that

  $$\begin{align*}\min(\mathcal{U})=\max(S), \enspace \max(\mathcal{U})=\max(T), \enspace S \cup \mathcal{U}\subseteq T, \end{align*}$$

  ordered by inclusion.

• • Composition is given by union.

These lax slice categories piece together into a 2-functor

so that, in particular for any $J\subset I$ with $i=\min (I)$ and $j=\min (J)$ , we have 2-functors

given on objects by the union of sets. It is an easy check that these functors are injective on objects, 1-morphisms, and 2-morphisms.

The 2-categories play a central role in our relative nerve construction.

Construction 4.3. Let

$$\begin{align*}F\colon {\mathbb{C}}^{\operatorname{op}}\to {\operatorname{Set}}_\Delta^{\mathbf{ms}} \end{align*}$$

be a ${\operatorname {Set}}_\Delta ^+$ -enriched functor. We define a marked-biscaled simplicial set as follows. An n-simplex consists of

• • A simplex $\sigma :\Delta ^n_\flat \to {\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})$ .

• • For every $\varnothing \neq I\subset [n]$ with $\min (I)=i$ , a map of marked-scaled simplicial sets

  

  such that, for every $\varnothing \neq J\subset I\subset [n]$ with $\min (J)=j$ and $\min (i)=i$ , the diagram

  

  commutes.

We then define markings and scalings on $\rho _{\mathbb {C}}(F)$ .

• • A 1-simplex is marked if the corresponding map descends to a map

  

• • A 2-simplex is lean if the corresponding map

  

  descends to a map

  

• • A 2-simplex is thin if and only if it is lean and the corresponding 2-simplex $\sigma :\Delta ^2 \to {\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})$ is thin.

Note that there is a canonical forgetful functor

which sends thin triangles to thin triangles.

Definition 4.4. The relative bicategorical nerve over a 2-category $\mathbb {C}$ is the functor

By the adjoint functor theorem, admits a left adjoint, which we will denote by

Lemma 4.5. The functor preserves trivial fibrations.

Proof. We need only check that the lifting problems

have solutions when $\mu :F\Rightarrow G$ is a projective (pointwise) trivial fibration and $f:A\to B$ is a generating cofibration of marked-biscaled simplicial sets. The proof is virtually identical to the proof of [3, Prop. 3.0.11].

Corollary 4.6. The functor preserves cofibrations.

4.1. Identifying

Let $\mathbb {C}$ be a 2-category. Then we can define a 2-functor

that maps a 1-morphism $f:c \to d$ to the functor given by precomposition with f. It is easy to verify that given a 2-morphism $\alpha :f \Rightarrow g$ we can construct a natural transformation $f^* \Rightarrow g^*$ whose component at an object $u:d \to x$ is given by $\alpha *u$ . Passing to ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories we thus obtain, for any strict 2-category $\mathbb {C}$ , a ${{\operatorname {Set}}_{\Delta }^+}$ -enriched functor

Definition 4.7. In the particular case where $\mathbb {C}={\mathbb {O}}^n$ , we will denote the functor constructed above by

$$\begin{align*}{\mathfrak{O}}^n\colon({\mathbb{O}}^n)^{\operatorname{op}} \to {\operatorname{Set}}_\Delta^{\mathbf{ms}} \end{align*}$$

Notation. The canonical normal lax functor $\xi _n:[n]\to {\mathbb {O}}^n$ gives rise to an inclusion of scaled simplicial sets which we denote by

$$\begin{align*}p_n\colon\Delta^n_\flat \to {\operatorname{N}}^{\operatorname{sc}}({\mathbb{O}}^n). \end{align*}$$

The map $p_n$ is also adjoint to the canonical map ${\mathfrak {C}}^{\operatorname {sc}}[\Delta ^n_\flat ]\to {\mathbb {O}}^n$ . We will equip $\Delta ^n$ with the minimal marking and lean scaling, and conventionally view $p_n$ as an object in $\left ({\operatorname {Set}}_\Delta ^{\mathbf {mb}}\right )_{/{\operatorname {N}}^{\operatorname {sc}}({\mathbb {O}}^n)}$ .

Lemma 4.8. Let $F\colon ({\mathbb {O}}^n)^{\operatorname {op}}\to {\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ be a ${\operatorname {Set}}_\Delta ^+$ -enriched functor. There is a natural bijection

Consequently, we have an equivalence of ${\operatorname {Set}}_\Delta ^+$ -enriched functors .

Proof. Follows immediately from unwinding the definitions.

Corollary 4.9. Denote by

$$\begin{align*}p_1^\sharp\colon (\Delta^1)^\sharp \to {\operatorname{N}}^{\operatorname{sc}}({\mathbb{O}}^1), \end{align*}$$

$$\begin{align*}(p_2)_{\flat\subset \sharp}\colon(\Delta^2)^\flat_{\flat\subset \sharp} \to {\operatorname{N}}^{\operatorname{sc}}({\mathbb{O}}^2), \text{ and} \end{align*}$$

$$\begin{align*}(p_2)_{\sharp}\colon(\Delta^2)^\flat_{\sharp} \to {\operatorname{N}}^{\operatorname{sc}}({\mathbb{O}}^2)_\sharp \end{align*}$$

the obvious decorated versions of the $p_n$ . Then

where $\dagger $ denotes the marking in which the unique morphism $02\to 012$ is marked.

Proof. All identifications except the last are immediate from the definitions. The additional marking in the final case follows from the necessity that the functor have source ${\mathbb {O}}^2_\sharp $ .

Notation. We will denote the three functors above by $({\mathfrak {O}}^1)^\sharp $ , $({\mathfrak {O}}^2)_{\flat \subset \sharp }$ and $(\mathfrak {O}_2)_\sharp $ , respectively.

4.2. Identifying $\mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^n}$

Our comparison will be with a very specific version of the straightening functor:

Notation. For a 2-category $\mathbb {C}$ , we view $\mathbb {C}$ as a ${\operatorname {Set}}_\Delta ^+$ -enriched category. The counit $\varepsilon _{\mathbb {C}}:{\mathfrak {C}}^{{\operatorname {sc}}}({\operatorname {N}}^{\operatorname {sc}}(\mathbb {C}))\to \mathbb {C}$ is an equivalence of ${\operatorname {Set}}_\Delta ^+$ -enriched categories. We will denote by

$$\begin{align*}\mathbb{S}\!{\operatorname{t}}_{\mathbb{C}}\colon({\operatorname{Set}}_{\Delta}^{\mathbf{mb}})_{/{\operatorname{N}}^{\operatorname{sc}}(\mathbb{C})}\to ({\operatorname{Set}}_\Delta^{\mathbf{ms}})^{{\mathbb{C}}^{\operatorname{op}}} \end{align*}$$

the relative straightening functor $\mathbb {S}\!{\operatorname {t}}_{\varepsilon _{\mathbb {C}}}$ .

We now unravel the definitions to characterise $\mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^n}(p_n)$ . By construction, the underlying functor to ${\operatorname {Set}}_\Delta ^+$ is given by the Yoneda embedding on the ${\operatorname {Set}}_\Delta ^+$ -enriched category

$$\begin{align*}{\mathbb{O}}^n \coprod_{{\mathfrak{C}}^{{\operatorname{sc}}}({\operatorname{N}}^{\operatorname{sc}}({\mathbb{O}}^n))} {\mathfrak{C}}^{{\operatorname{sc}}}({\operatorname{N}}^{\operatorname{sc}}({\mathbb{O}}^n)) \coprod_{{\mathfrak{C}}^{{\operatorname{sc}}}(\Delta^n_\flat)} {\mathfrak{C}}^{{\operatorname{sc}}}((\Delta^n_\flat)^\triangleright) \end{align*}$$

We note that ${\mathbb {O}}^n={\mathfrak {C}}^{{\operatorname {sc}}}(\Delta ^n_\flat )$ , and by the triangle identities for the adjunction ${\mathfrak {C}}^{{\operatorname {sc}}}\dashv {\operatorname {N}}^{\operatorname {sc}}$ , we see that the induced map

$$\begin{align*}{\mathfrak{C}}^{{\operatorname{sc}}}(\Delta^n_\flat)\to {\mathbb{O}}^n \end{align*}$$

is simply the identity. The pushout above thus collapses to simply ${\mathfrak {C}}^{\operatorname {sc}}((\Delta ^n_\flat )^\triangleright )$ . We can then describe the marked-scaled simplicial set $\mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^n}(\Delta ^n_\flat )(i)$ (up to isomorphism) as the poset $\mathcal {L}^n_\flat (i)$ given in Definition 3.11.

Construction 4.10. We construct a morphism of marked-scaled simplicial sets whose underlying map of simplicial sets is given by (the nerve of) a normal lax functor defined as follows:

• • On objects, $S\mapsto S$ .

• • On morphisms $S\subset T$ is sent to the morphism $\{\max (S),\max (T)\}:S\to T$ .

The fact that, for $S\subset T\subset V$ , we have $\{\max (S),\max (V)\}\subset \{\max (S),\max (T),\max (V)\}$ gives us our compositors. The fact that if $S=T$ , we have $\{\max (S),\max (T)\}=\{\max (S)\}$ gives strict unitality. Since both marked-scaled simplicial sets carry the minimal marking we only need to check that $\eta ^n_i$ preserves the scaling. Let $S_0\subset S_1\subset S_2$ be a 2-simplex in the source. If there are $i,j$ such that $\max (S_i)=\max (S_j)$ , then it follows immediately that $\{\max (S_0),\max (S_1),\max (S_2)\}=\{\max (S_0),\max (S_2)\}$ .

The following lemma follows immediately from our definitions.

Lemma 4.11. The maps $\eta ^n_i$ define natural transformations of ${\operatorname {Set}}_\Delta ^+$ -enriched functors $\eta ^n:\mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^n}(\Delta ^n_\flat )\to {\mathfrak {O}}^n$ .

Proposition 4.12. The morphisms $\eta _i^n: \mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^n}(\Delta ^n_\flat )(i)\to {\mathfrak {O}}^n(i)$ are equivalences of marked-scaled simplicial sets.

We will prove this proposition in a series of lemmata. Since both simplicial sets are equipped with the minimal marking, it suffices to show that the map is an equivalence on underlying scaled simplicial sets by Theorem 2.52. Since , it suffices to show that the induced map

is an equivalence of ${\operatorname {Set}}_\Delta ^+$ -enriched categories. Since this map is clearly bijective on objects, it suffices to check that the induced morphisms on mapping spaces are equivalences.

In both cases, the mapping spaces are nerves of posets.

• • For , the mapping space ${\mathfrak {O}}^n_i(S,T)$ is the poset of chains $U\subset [n]$ such that $\min (U)=\max (S)$ , $\max (U)=\max (T)$ , and $S\cup U\subset T$ . Equivalently, this is the poset ${\mathbb {O}}^T(\max (S),\max (T))$ , equipped with the minimal marking.

• • For $S,T\in Q^n_i$ , the mapping space ${\mathfrak {C}}^{{\operatorname {sc}}}[(\mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^n}(\Delta ^n_\flat ))(i)](S,T)$ is the poset of chains

  $$\begin{align*}S\subset S_1\subset \cdots \subset S_k\subset T \end{align*}$$

  in $\mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^n}(\Delta ^n_\flat )(i)$ . An inclusion $\vec {S}\subset \vec {U}$ is marked if and only if, for every $S_i\subset S_{i+1}$ in $\vec {S}$ , every $U_\ell \in \vec {U}$ between $S_i$ and $S_{i+1}$ , either $\max (U_\ell )=\max (S_i)$ or $\max (U_\ell )=\max (S_{i+1})$ . Notice that could have elements lower than $\max (S)$ .

The map

$$\begin{align*}\xi^n_i: {\mathfrak{C}}^{{\operatorname{sc}}}[(\mathbb{S}\!{\operatorname{t}}_{{\mathbb{O}}^n}(\Delta^n_\flat))(i)](S,T) \to {\mathfrak{O}}^n_i(S,T) \end{align*}$$

sends a chain $S\subset S_1\subset \cdots \subset S_k\subset T$ to the chain

$$\begin{align*}\xi^n_i(\vec{S})=\bigcup_{V\in \vec{S}} \{\max(V)\}. \end{align*}$$

Definition 4.13. For ease of notation, we define

$$\begin{align*}\mathbb{L}^n_i:={\mathfrak{C}}^{{\operatorname{sc}}}[(\mathbb{S}\!{\operatorname{t}}_{{\mathbb{O}}^n}(\Delta^n_\flat))(i)] \end{align*}$$

For any $S,T\in \mathbb {L}^n_i$ , we define a full subposet $\mathbb {B}^{n}_i(S,T)\subset \mathbb {L}^n_i(S,T)$ consisting of chains

$$\begin{align*}S\subset [T,s_1]\subset \cdots \subset [T,s_k]\subset T \end{align*}$$

where we define for every $s \in T$ the subset .

Lemma 4.14. An inclusion $\vec {S}\subset \vec {U}$ represents a marked morphism in $\mathbb {L}^n_i(S,T)$ if and only if its image under $\xi ^n_i$ is degenerate.

Proof. Immediate from the definition.

Lemma 4.15. The restriction of the map $\xi ^{n}_i$

$$\begin{align*}\xi^n_i\colon\mathbb{B}^n_i(S,T)\to {\mathbb{O}}^T(\max(S),\max(T)) \end{align*}$$

is an equivalence of marked simplicial sets.

Proof. We define a map

$$\begin{align*}\gamma\colon{\mathbb{O}}^T(\max(S),\max(T))\to \mathbb{B}^n_i(S,T) \end{align*}$$

which sends $\max (S)<s_1<\cdots <s_k<\max (T)$ to the chain

$$\begin{align*}S\subset [T,s_1]\subset \cdots \subset [T,s_k]\subset T. \end{align*}$$

We then note that $\xi ^n_i\circ \gamma =\operatorname {id}$ . We claim that $\gamma \circ \xi ^n_i\leqslant \operatorname {id}$ , which yields a marked homotopy $\gamma \circ \xi ^n_i$ to $\operatorname {id}$ . To prove the claim we note that $\gamma \circ \xi ^n_i(\vec {S})$ is given by $\vec {S}$ if $S_1 \neq [T,s_0]$ with $s_0=\max (S)$ or by in which case the existence of the marked morphism $\gamma \circ \xi ^n_i(\vec {S}) \to \vec {S}$ follows immediately.

Lemma 4.16. The inclusion $\iota :\mathbb {B}^n_i(S,T)\to \mathbb {L}^n_i(S,T)$ is an equivalence of marked simplicial sets.

Proof. Let $s_j \in T$ . We define $\mathbb {L}^s \subset \mathbb {L}^n_i(S,T)$ as the full subposet consisting of those chains

$$\begin{align*}\vec{S}=S \subset S_1 \subset \cdots \subset S_k \subset T, \end{align*}$$

such that $S_i=[T,s_i]$ whenever $s_i \geqslant s_j$ . Note that if $s_j\leqslant s_0=\max (S)$ then it follows that $\mathbb {L}^s=\mathbb {B}^n_i(S,T) $ . Consider a filtration

$$\begin{align*}\mathbb{B}^n_i(S,T)=\mathbb{L}^{s_0} \subset \mathbb{L}^{s_1}\subset \cdots \subset \mathbb{L}^{s_m} \subset \mathbb{L}^{s_m+1}=\mathbb{L}^n_i(S,T), \enspace \text{ with }s_m=\max(T) \end{align*}$$

Our goal is to show that each step in the filtration is a weak equivalence of marked simplicial sets. We denote by $\iota _j: \mathbb {L}^{j} \to \mathbb {L}^{j+1}$ for $j=0,\dots ,s_m$ . Let $\vec {S}=S \subset S_1 \subset \cdots \subset S_k \subset T$ be an object of $\mathbb {L}^{j+1}$ we construct a new chain $\pi _j(\vec {S})$ by replacing each $S_\ell $ with $s_{\ell } \geqslant s_{j}$ with its corresponding $[T,s_\ell ]$ . This definition yields a functor

$$\begin{align*}\pi_j\colon \mathbb{L}^{j+1} \to \mathbb{L}^j; \vec{S} \mapsto \pi_j(\vec{S}) \end{align*}$$

such that $\pi _j \circ \iota _j=\operatorname {id}$ . Let $\zeta _j=\iota _j \circ \pi _j$ . We construct a functor

$$\begin{align*}\theta_j\colon\mathbb{L}^{j+1} \to \mathbb{L}^{j+1} \end{align*}$$

that appends to each chain $\vec {S} \in \mathbb {L}^{j+1}$ the object $[T,s_j]$ if there exists some $S_\ell \in \vec {S}$ such that $\max (S_{\ell })=s_j$ or leaves the chain untouched otherwise. Note that if $s_j=s_m$ then this functor is the identity. We also observe that we have a natural transformation $\operatorname {id} \leqslant \theta _j$ and $\zeta _j \leqslant \theta _j$ whose components are marked. It follows that each $\iota _j$ is a weak equivalence and consequently so is $\iota $ .

Proof of Proposition 4.12.

We simply apply Lemma 4.16, Lemma 4.15, and 2-out-of-3.

Turning now to the cases $(p_1)^\sharp $ , $(p_2)_{\flat \subset \sharp }$ and $(p_2)_\sharp $ , we see that the corresponding straightenings are obtained from $\mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^1}(p_1)$ and $\mathbb {S}\!{\operatorname {t}}_{{\mathbb {O}}^2}(p_2)$ by maximally marking or maximally scaling the values of the functors, respectively. We then have the following

Corollary 4.17. The transformations $\xi ^n$ , $n=1,2$ induce equivalences of enriched functors

$$\begin{align*}(\xi^1)^\sharp\colon\mathbb{S}\!{\operatorname{t}}_{{\mathbb{O}}^1}(p_1^\sharp)\to ({\mathfrak{O}}^1)^\sharp, \end{align*}$$

$$\begin{align*}(\xi^2)_{\flat\subset \sharp}\colon\mathbb{S}\!{\operatorname{t}}_{{\mathbb{O}}^2}((p_2)_{\flat\subset \sharp})\to ({\mathfrak{O}}^2)_{\flat\subset\sharp}, \text{ and} \end{align*}$$

$$\begin{align*}(\xi^2)_{ \sharp}\colon\mathbb{S}\!{\operatorname{t}}_{{\mathbb{O}}^2_\sharp}((p_2)_{ \sharp})\to ({\mathfrak{O}}^2)_{\sharp} \end{align*}$$

Proof. The morphism $(\xi ^1)^\sharp $ is an isomorphism, and it is a quick check to extend the previous arguments to cover the case $(\xi ^2)_{\flat \subset \sharp }$ . One then notes that, for each $i\in {\mathbb {O}}^2$ , the i-component of $(\xi ^2)_\sharp $ is a pushout of the i-component of $(\xi ^2)_{\flat \subset \sharp }$ along the inclusion $(\Delta ^1)^\flat \to (\Delta ^1)^\sharp $ , and thus is an equivalence.

Remark 4.18. As in [4, Prop. 4.1.1] any 2-functor $f:\mathbb {C}\to \mathbb {D}$ yields diagrams

and

which commute up to natural isomorphism.

Theorem 4.19. There exists a unique family of natural weak equivalences

indexed by pairs $(\mathbb {C},X)$ consisting of a 2-category $\mathbb {C}$ and $X\in ({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/{\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})}$ with the following properties.

1. 1. On the maps $p_n$ for $n\geqslant 0$ , $p_1^\sharp $ , $(p_2)_{\flat \subset \sharp }$ and $(p_2)_\sharp $ , the transformations $\xi ^{\mathbb {C}}(X)$ coincide with the transformations $\xi ^n$ from Proposition 4.12 and Corollary 4.17.

2. 2. For every map $g:X\to Y$ in $({\operatorname {Set}}_{\Delta }^{\mathbf {mb}})_{/{\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})}$ , the diagram

   

   commutes

3. 3. For every 2-functor $f:\mathbb {C}\to \mathbb {D}$ , the diagram

   

   commutes.

Proof. This is identical to the proofs of [4, Prop 4.3.1 and 4.3.2].

Corollary 4.20. The adjunction

is a Quillen equivalence.

Proof. Since preserves cofibrations, and is naturally weakly equivalent to $\mathbb {S}\!{\operatorname {t}}_{\mathbb {C}}$ , it preserves trivial cofibrations, and thus is left Quillen. Moreover, the left-derived functors of $\mathbb {S}\!{\operatorname {t}}_{\mathbb {C}}$ and agree, and the former is an equivalence.

4.3. Comparison to the strict case

We now establish a comparison result with the strict 2-categorical case, as worked out by Buckley in [8]. We will heavily use the fact that the Duskin 2-nerve ${\operatorname {N}}_2(\mathbb {C})$ of any strict 2-category $\mathbb {C}$ is 3-coskeletal, which will allow us to construct a comparison map by checking a finite number of cases by hand. Once the comparison is established, we will show that this map is fibre-wise equivalence of $\infty $ -bicategories.

Let us now introduce the 2-categorical Grothendieck construction we wish to compare with. Appropriately dualising Buckley’s construction,⁵ the strict 2-categorical Grothendieck construction of a 2-functor

$$\begin{align*}F\colon{\mathbb{C}}^{\operatorname{op}}\to 2\!\operatorname{Cat} \end{align*}$$

is the 2-category $\operatorname {El}(F)$ which has

• • Objects: pairs $(x,x_-)$ with $x\in \mathbb {C}$ and $x_-\in F(x)$ .

• • Morphisms:

  $$\begin{align*}(f,f_-)\colon(x,x_-)\to (y,y_-) \end{align*}$$

  where $f:x\to y$ in $\mathbb {C}$ , and $f_-:x_-\to F(f)(y_-)$ .

• • 2-Morphisms: $(\alpha ,\alpha _-):(f,f_-)\Rightarrow (g,g_-)$ , where $\alpha :f\Rightarrow g$ is a 2-morphism in $\mathbb {C}$ , and $\alpha _-$ fits in the diagram

  

The resulting functor $\operatorname {El}(F)\to \mathbb {C}$ is a 2-Cartesian fibration, where

• • $(f,f_-)$ is Cartesian if $f_-$ is an equivalence, and

• • $(\alpha ,\alpha _-)$ is coCartesian if $\alpha _-$ is an isomorphism.

Our aim is to prove the following

Theorem 4.21. Let

$$\begin{align*}F\colon{\mathbb{C}}^{({\operatorname{op}},-)}\to 2\!\operatorname{Cat} \end{align*}$$

be a 2-functor. Then there is an equivalence

of 2-Cartesian fibrations over ${\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})$ .

To construct the map, we first formulate a description of ${\operatorname {N}}^{\operatorname {sc}}(\operatorname {El}(F))$ .

Construction 4.22. We now define a strict version of , which we will show is isomorphic to $\operatorname {El}(F)$ .

Let

$$\begin{align*}F\colon {\mathbb{C}}^{\operatorname{op}}\to 2\operatorname{Cat} \end{align*}$$

2-functor. We define a marked-biscaled simplicial set $\Psi _{\mathbb {C}}(F)$ as follows. An n-simplex $\Delta ^n\to \Psi _{\mathbb {C}}(F)$ consists of

• • A 2-functor $\sigma \colon {\mathbb {O}}^n\to \mathbb {C}$ .

• • For every $\varnothing \neq I\subset [n]$ with $\min (I)=i$ , a functor of 2-categories

  

  such that, for every $\varnothing \neq J\subset I\subset [n]$ with $\min (J)=j$ and $\min (i)=i$ , the diagram

  

  commutes.

We equip this simplicial set with markings and scalings exactly as in Construction 4.3.

Lemma 4.23. There is an isomorphism of scaled simplicial sets over ${\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})$

$$\begin{align*}{\operatorname{N}}^{\operatorname{sc}}(\operatorname{El}(F))\cong \Psi_{\mathbb{C}}(F). \end{align*}$$

Proof. By the 3-coskeletalness of the Duskin nerve, it suffices to provide an isomorphism on 3-truncated simplicial sets.

On 0- and 1-simplices, the data specified by the simplices in both constructions is identical. A 2-simplex in $\Psi _{\mathbb {C}}(F)$ consists of the following data:

• • A 2-simplex in $\mathbb {C}$

  

• • Three 1-simplices

  $$\begin{align*}f_{12}\colon x^1_-\to \phi_{12}^\ast(x^2_-) \end{align*}$$
  $$\begin{align*}f_{01} \colon x^0_-\to \phi_{01}^\ast(x^1_-) \end{align*}$$

  and

  $$\begin{align*}f_{02}\colon x^0_-\to \phi_{02}^\ast(x^2_-) \end{align*}$$

• • A diagram

  

  in $F(0)$ .

These data are identical to the data of a 2-simplex in $\operatorname {El}(F)$ . It is immediate that scaled 2-simplices coincide under this correspondence.

Finally, we note that 3-simplices in $\operatorname {El}(F)$ are simply compatibility conditions on 2-morphisms. It is a long but easy check to see that, given a 3-simplex in $\Psi _{\mathbb {C}}(F)$ , the corresponding 2-simplices in $\operatorname {El}(F)$ are compatible and vice versa.

Construction 4.24. We construct a map

using the description of Lemma 4.23 by sending the simplex defined by $\sigma : {\mathbb {O}}^n\to \mathbb {C}$ and to the simplex of defined by the nerves of $\sigma $ and the $\theta _I$ . It is easy to check that $\tau $ preserves Cartesian edges and coCartesian triangles over ${\operatorname {N}}^{\operatorname {sc}}(\mathbb {C})$ .

Definition 4.25. We denote by the canonical projection. This sends $S\mapsto \max (S)$ , and sends a morphism to the set .

Given a 2-category , we call a 2-functor

peripatetically constant if it sends every morphism where $\max (S)=\max (T)$ to an identity, and every 2-morphism between such morphisms to an identity as well. We denote the set of peripatetically constant functors by

Proposition 4.26. Let $\mathbb {D}$ be a 2-category, and denote by $\ast $ the terminal 2-category. There is a weak equivalence of scaled simplicial sets,

Proof. We claim that restriction along $\pi ^n$ (cf. Definition 4.25) induces a weak equivalence . Note that we for $n \geqslant 0$ we have a commutative diagram

where the top horizontal functor is a weak equivalence by Lemma 3.38, the left-most vertical map is an equivalence by Theorem 4.19 and the right-most vertical map is an equivalence by Theorem 2.14. By 2-out-of-3 the bottom horizontal map is also a weak equivalence. Let us remark that this discussion holds mutatis mutandis for the remaining generators of ${\operatorname {Set}}_\Delta ^{\mathbf {ms}}$ . We conclude that after passing to adjoints we have a commutative diagram

where all morphisms involved are weak equivalences. There is a section ${\operatorname {N}}^{\operatorname {sc}}(\mathbb {D}) \to \widetilde {{\operatorname {N}}^{\operatorname {sc}}(\mathbb {D})}$ which sends a simplex $\Delta ^n_\flat \to {\operatorname {N}}^{\operatorname {sc}}(\mathbb {D})$ represented by a 2-functor $f \colon {\mathbb {O}}^n \to \mathbb {D}$ to the functor of scaled simplicial sets obtained via applying ${\operatorname {N}}^{\operatorname {sc}}(-)$ . We can identify the resulting map with $\pi ^*$ thus proving the claim.

Lemma 4.27. The 2-functors $\pi ^n$ induce bijections

where the latter denotes the set of peripatetically constant functors. In particularly, restricting strict 2-functors along the $\pi ^n$ induces an isomorphism ${\operatorname {N}}^{\operatorname {sc}}(F(x))\cong {\operatorname {N}}^{\operatorname {sc}}(\operatorname {El}(F|_{\{x\}}))$ .

Proof. We can define a strict 2-functor

which acts as the identity on 1- and 2-morphisms. Since $\pi ^n\circ s^n=\operatorname {id}$ , we have $(s^n)^\ast \circ (\pi ^n)^\ast =\operatorname {id}$ , and thus, $\pi ^n_\ast $ is injective. It is immediate from unraveling the definitions that the image of $(\pi ^n)^\ast $ is precisely the peripatetically constant functors.

Proof of Theorem 4.21.

By [5, Proposition 3.35], it will suffice for us to show that $\tau $ is an equivalence on fibres. By construction, the fibre of ${\operatorname {N}}^{\operatorname {sc}}(\operatorname {El}(F))$ over $x\in \mathbb {C}$ is precisely ${\operatorname {N}}^{\operatorname {sc}}(\operatorname {El}(F|_{\{x\}}))$ , and the fibre of over x is precisely .

By construction, we have a commutative diagram

where, in a slight abuse of notation, the top-left functor is restriction of strict 2-functors along and the bottom functor is restriction along the nerves ${\operatorname {N}}^{\operatorname {sc}}(\pi ^n)$ . By Lemma 4.27, the top left morphism is an isomorphism, and by Proposition 4.26, the bottom morphism is an equivalence. Thus, $\tau $ is a fibrewise equivalence, completing the proof.