FIRST-ORDER MODAL LOGIC VIA LOGICAL CATEGORIES

Abstract

We extend the logical categories framework to first-order modal logic. In our modal categories, modal operators are applied directly to subobjects and interact with the background factorization system. We prove a Joyal-style representation theorem into relational structures formalizing a ‘counterpart’ notion. We investigate saturation conditions related to definability questions and we enrich our framework with quotients and disjoint sums, thus leading to the notion of a modal (quasi) pretopos. We finally show a way to build syntactic categories out of first-order modal theories.

1 Introduction

Modal logic is a pervasive formalism with applications in computer science, mathematics, and philosophy. The core modal system is the minimal normal modal logic system (called K) which is axiomatized at the propositional level by adding to classical tautologies and modus ponens the distribution law

(1) $$ \begin{align} \Box (\phi \to \psi) \to (\Box \phi \to \Box \psi) \end{align} $$

and the necessitation rule

(2) $$ \begin{align} \mathrm{from}~~~\phi~~~~~\mathrm{infer}~~~~~\Box \phi. \end{align} $$

Algebraically, this axiomatization corresponds (via standard Lindenbaum constructions) to modal algebras which are Boolean algebras endowed with a unary operator subject to the equations

$$ \begin{align*}\Box (x\wedge y) = \Box x \wedge \Box y \qquad \Box \top =\top. \end{align*} $$

Using equivalently the dual ‘possibility’ operator $\Diamond := \neg \Box \neg $ such equations can be rewritten as

$$ \begin{align*}\Diamond (x\vee y) = \Diamond x \vee \Diamond y \qquad \Diamond \bot =\bot. \end{align*} $$

In the applications, basic modal operators satisfying the axioms of K serve as building blocks for more complex modalities obtained by combining them with each other and by applying various kinds of constructors, typically fixpoint constructors. In this way, many logics like PDL [11], LTL, CTL, and CTL $^*$ [26] (culminating in the $\mu $ -calculus [7]) arose in the formal verification area. Similarly, in the knowledge representation area, basic K-operators are combined to give rise to various description logics like $\mathcal {ALC}$ and extensions [3].

In mathematical frameworks of a topological nature, the typical modal system is the system $S4$ obtained by adding to K the further ‘reflexivity’ and ‘transitivity’ axioms

$$ \begin{align*}\Box \phi \to \phi, \qquad \Box \phi \to \Box \Box \phi; \end{align*} $$

such axioms correspond, at the algebraic level, to interior algebras, i.e., to modal algebras whose $\Box $ operator satisfies precisely the axioms of the interior operator in topological spaces [37].

Most of the literature in modal logic is confined to propositional systems. However, predicate extensions turn out to be important too, e.g.: (i) in the formal verification area, where relative completeness [36] brings together temporal reasoning and decision procedures and (ii) in the knowledge representation area, where decidability results can surprisingly be obtained for some expressive fragments of predicate modal logics, like the so-called “monodic” fragments [8, 12].

However, a categorical approach via logical categories ¹ to quantified modal logic has to face the structural problem of the lack of homogeneity between interpretation of sorts and predicates that arises in standard semantics like Kripke semantics. The problem can be explained as follows. A predicate Kripke frame [13, 29] is a triple $(W,R, D)$ , where W is a set, $R\subseteq W\times W$ is a relation, and D is a function associating a set $D_w$ with every $w\in W$ , with the condition $D_w\subseteq D_v$ in case $wRv$ . In a predicate Kripke frame, an n-ary relation symbol P is interpreted as a collection of subsets $P_w\subseteq D_w^n$ indexed by $w\in W$ , but nothing forces $P_w \subseteq P_v$ when $wRv$ . Thus, contrary to what happens for intuitionistic logic, the interpretation of a relation symbol is not a predicate Kripke frame itself and this is incompatible with the categorical logic practice of interpreting relation symbols (and more generally formulae) as subobjects.

A possible way out would be the adoption of Lawvere doctrines [30] as a categorical framework: a modal Lawvere doctrine could be easily defined (see, e.g., [42]) as a pair given by a category with products and a contravariant functor with values in the category of modal algebras (suitable conditions like the Beck–Chevalley condition should also be added). However, this solution (if not accompanied by suitable comprehension principles as in [31]) would just certify the above lack of homogeneity between the interpretation of terms and formulae; in addition, it would contradict one of the main principles of the categorical approach to logic, namely presentation independence. After all, whether a certain definable set is logically handled using a sort symbol or a formula is a question of presentation (analogously, whether a definable function is mirrored by a term or by a functional relation is also a matter of presentation). Unfortunately, it is precisely comprehension which fails in the above-mentioned semantic modal contexts.

A more intriguing solution was proposed by F. W. Lawvere and implemented in [41]. The asymmetry between sorts and predicates is conceptually explained by saying that our semantics involves in fact two toposes and a geometric morphism connecting them: from the first topos, one takes the interpretation of sorts and function symbols and from the other one, one takes the interpretation of relation symbols and formulae. This schema encompasses as a special case Kripke semantics, its generalization to presheaves [15, 17] and also the interpretation of the modality as interior in étale spaces [1, 2, 8]. A representation theorem in Joyal style [34] for pairs of logical categories connected by a functor having local adjoints was proved in [35].

This solution has the major merit of not introducing modalities as extra features, but of generating them out of the pure basic category-theoretic machinery, as it happens with all the classical and intuitionistic logical operators. Still, the solution suffers from a considerable drawback: $S4$ axioms are forced whenever modalities are produced by applying the above schema involving two different categories connected by a functor. This is unsatisfactory because, as pointed out above, the weaker system K seems to be involved in most applications outside the realm of pure mathematics. However, precisely turning back to a pure mathematical context, another way out appears. In topological models, modalities (namely the interior or its dual closure operator) apply to subsets, aka subspaces. Subspaces are regular subobjects, i.e., subsets with the induced topology. This simple observation reveals an unexpected solution, whose main feature is that of changing the underlying factorization system from the regular epi/mono factorization system adopted in logical categories, to an abstract factorization system, satisfying minimal conditions (like pullback stability).

One may reach the same conclusion by generalizing Kripke semantics too. A long-debated question among philosophers [25, 32] is the trans-world re-identification problem. In Kripke semantics, the question is handled trivially: it is assumed that $D_w\subseteq D_v$ whenever we have $wRv$ . This means that some $a\in D_w$ is identified with itself in all the accessible worlds v. However, this is no longer true if we generalize predicate Kripke frames to presheaves, because there the transition from $D_w$ to $D_v$ takes the form of a function or even of a multiplicity of functions, if presheaves are taken over a category which is not a poset. Once the Kripkean inclusion trivialization is broken, nothing prevents from taking relations as transitions from a domain $D_v$ to a domain $D_w$ : such relations are called ‘counterpart relations.’ In the resulting semantics, the forcing condition for $\Diamond $ becomes “ $a\in D_w$ satisfies $\Diamond P$ iff some counterpart $b\in D_v$ of a (with $w R v$ ) satisfies P” (see the defining condition (9) in Section 3 below). A formal realization of this semantics was introduced in [15]; the universes generalizing predicate Kripke frames and presheaves were there called ‘relational presheaves’ and further analyzed, e.g., in [8, 14, 38, 39]. What we want to point out now is that the asymmetry between interpretation of sorts and predicates disappears in such relational presheaves, because a ‘collection of subsets indexed by possible worlds’ is, once again, just a regular mono (this is because relations, unlike inclusions or functions, can be restricted to arbitrary subsets).

In Section 3 we take up relational presheaves as our fundamental semantic environment; however we shall speak of ‘relational $\mathbf {G}$ -sets’ instead, because we assume the variation domain to be a graph $\mathbf {G}$ and not a category. This choice was already suggested in [15] and is needed in order to avoid the validity of $S4$ axioms; moreover, we shall see that we get the surprisingly much richer categorical structure of a quasi-topos (see Proposition 3.2 below) by adopting relational $\mathbf {G}$ -sets instead of relational presheaves, which motivates once more the relevance of K modalities.

The article is structured as follows: in Section 2 we review some preliminary material on factorization systems, in Section 3 we introduce modal categories and supply examples of them. Then we prove a representation theorem in Joyal style for modal categories into relational $\mathbf {G}$ -sets (Section 4). The article continues with a detailed analysis of saturation aspects in modal categories (Section 5) and then it extends the representation theorem to quotients and disjoint unions, i.e., to modal quasi-pretoposes (Section 6). Lastly, in Section 7 we discuss a construction of syntactic categories out of modal first-order theories. We conclude the article with a discussion on further developments (Section 8).

Notations. In this article, the composition of two arrows $f : X\longrightarrow Y$ and ${g : Y\longrightarrow Z}$ is written $fg$ or $g\circ f$ . When we speak of semilattices, we always mean meet-semilattices with unit and when we speak of lattices we always mean lattices with unit and zero. A left exact or lex category is a category with finite limits. A lex functor is a functor preserving finite limits.

2 Background on factorization systems

Coherent categories and pretoposes have been thoroughly studied as the syntactic categories of first-order coherent theories. Several logics have been given a similar categorical presentation. Among the most satisfying examples, we can mention geometric logic with Grothendieck toposes, higher-order intuitionistic logic with elementary toposes, and intuitionistic logic with Heyting pretoposes. We refer the interested reader to [27, Section D1] or [34].

We will define in Section 3 modal categories in a similar spirit, as categories in which first-order modal logic can be interpreted. When X is an object of a coherent category, the ‘propositions’ in context X are all the monomorphisms with codomain X. For modal logic, however, we need to add some extra structure. First, we need to specify which monomorphisms should be interpreted as ‘propositions.’ For instance a ‘proposition’ on a topological space should be a subset of the underlying set, hence a subspace, and not all the monomorphisms are subspace embeddings. Second, we need to specify how the modalities act on these propositions/subspaces.

In order to account for the fact that not all the monomorphisms are treated as ‘subspaces,’ we need to relax the usual categorical notions of regular and coherent categories. Yet, we will maintain the important property that every morphism factors as a ‘surjection’ followed by a ‘subspace embedding.’ Hence, we start with orthogonal factorization systems.

2.1 Factorization systems and f-regular categories

We will recall some standard facts about orthogonal factorization systems and we refer the reader to [22, Section 3.1] and [6, Section 5.5] for more details.

Let ${\mathbf {E}}$ be a lex category. An orthogonal factorization system on ${\mathbf {E}}$ can be defined as a pair $({\mathcal {E}}, {\mathcal {M}})$ of classes of morphisms of ${\mathbf {E}}$ such that:

• • Each morphism of ${\mathbf {E}}$ can be written as $em$ with $e \in {\mathcal {E}}$ and $m \in {\mathcal {M}}$ .

• • Both ${\mathcal {E}}$ and ${\mathcal {M}}$ contain identities and are closed under left and right composition with isomorphisms.

• • Every arrow of ${\mathcal {E}}$ is orthogonal on the left to every arrow of ${\mathcal {M}}$ . In other words, for each commutative square,

  
  where $e \in {\mathcal {E}}$ and $m \in {\mathcal {M}}$ , there is a unique diagonal filler $w : B\longrightarrow C$ such that $ew = u$ and $wm = v$ .

In an orthogonal factorization system, the factorization of a morphism as $e\,m$ with $e \in {\mathcal {E}}$ and $m \in {\mathcal {M}}$ is unique up to a unique isomorphism, and both ${\mathcal {E}}$ and ${\mathcal {M}}$ are stable under composition.

In order to interpret first-order regular logic in ${\mathbf {E}}$ , we need to impose some further conditions. The most essential one is stability: An orthogonal factorization system $({\mathcal {E}}, {\mathcal {M}})$ is called stable if ${\mathcal {E}}$ and ${\mathcal {M}}$ are stable under pullbacks. (In fact, ${\mathcal {M}}$ is necessarily stable under pullbacks in an orthogonal factorization system.)

Definition 2.1. An f-regular category (short for ‘factorization-regular category’) is a lex category ${\mathbf {E}}$ equipped with a stable orthogonal factorization system $({\mathcal {E}}, {\mathcal {M}})$ such that ${\mathcal {M}}$ is included in the class of all monomorphisms, and contains the class of regular monomorphisms. A lex functor between f-regular categories is f-regular when it preserves the factorization system.

In Definition 2.1, the condition that all the arrows in ${\mathcal {M}}$ are monomorphic ensures that we can think of them as subobjects. As we will see in Section 2.2, this allows us to interpret regular first-order logic in ${\mathbf {E}}$ by means of a Lawvere doctrine. In terms of this doctrine, the additional requirement that every regular monomorphism is in ${\mathcal {M}}$ means that if two parallel arrows f and g satisfy $\vdash f(x)=g(x)$ in the internal language, then $f=g$ . As shown in Lemma 2.2, this requirement admits several reformulations using either regular monomorphisms, diagonal maps, or pairs.

Lemma 2.2. Let ${\mathbf {E}}$ be a lex category equipped with a stable orthogonal factorization system $({\mathcal {E}}, {\mathcal {M}})$ . Then the following are equivalent:

1. 1. ${\mathcal {M}}$ contains the regular monomorphisms;

2. 2. ${\mathcal {M}}$ contains the diagonals $\Delta _X:X\longrightarrow X\times X$ ;

3. 3. if $f:X\longrightarrow Y$ and $g:X\longrightarrow Z$ are in ${\mathcal {M}}$ , then $(f,g):X\longrightarrow Y\times Z$ is in ${\mathcal {M}}$ .

Proof. Suppose 1. Then ${\mathcal {M}}$ contains the diagonals since they are regular monomorphisms. Reciprocally, suppose 2 and let $f,g:X\rightrightarrows Y$ . The equalizer of f and g is also the pullback of $\Delta _Y:Y\longrightarrow Y\times Y$ by $(f,g):X\longrightarrow Y\times Y$ , hence it is in ${\mathcal {M}}$ .

Suppose again 2 and let $f:X\longrightarrow Y$ and $g:X\longrightarrow Z$ be in ${\mathcal {M}}$ . By pullback stability, $f\times 1_X : X\times X\longrightarrow Y\times X$ and $1_Y\times g : Y\times X\longrightarrow Y\times Z$ are in ${\mathcal {M}}$ . We obtain by composing that $(f,g) = \Delta _X (f\times 1_X) (1_Y\times g)$ is in ${\mathcal {M}}$ , hence 3 is satisfied. Reciprocally, 2 is a special case of 3 with $f = g = 1_X$ .

Since ${\mathcal {M}}$ suffices to determine ${\mathcal {E}}$ (because ${\mathcal {E}}$ turns out to be the left orthogonal class ${\mathcal {M}}^\bot $ to ${\mathcal {M}}$ ), we will denote in general an f-regular category by the pair $({\mathbf {E}}, {\mathcal {M}})$ , leaving ${\mathcal {E}} = {\mathcal {M}}^\bot $ implicit. Sometimes ${\mathcal {M}}$ too will be left implicit. Every regular category is f-regular, with the factorization system of regular epimorphisms and monomorphisms. An r-regular category, as defined in [22, Section 3.1], is an f-regular category whose factorization system consists of epimorphisms and regular monomorphisms. For instance, the category $\mathbf {Top}$ of topological spaces and continuous functions is r-regular.

The following proposition shows that in an f-regular category, the left class ${\mathcal {E}}$ satisfies the dual properties of the class $ \mathcal {M}$ .

Proposition 2.3. In an f-regular category with factorization system $({\mathcal {E}}, \mathcal {M})$ , we have that ${\mathcal {E}}$ includes regular epis and is included in the class of all epis.

Proof. Let p be the coequalizer of $g_1,g_2$ and let $m\circ q$ be an $({\mathcal {E}}, \mathcal {M})$ -factorization of p. By the universal property of coequalizers there is h such that $h\circ p=q$ . Then $m\circ h\circ p=m\circ q= p$ and so $m\circ h=\mathrm {id}$ because p is epi being a coequalizer. Now m is iso because it is mono (recall $m\in \mathcal {M}$ ) and split epi. This shows that $p\in {\mathcal {E}}$ .

Let now $p\in {\mathcal {E}}$ be such that $g_1\circ p = g_2\circ p$ and take the equalizer m of $g_1,g_2$ . Then there is h such that $m\circ h=p$ . Since $m\in \mathcal {M}$ and m is orthogonal to p there is l such that $m\circ l=id$ . But then m is iso (being split epi and mono), which means that $g_1=g_2$ .

2.2 The Lawvere doctrine of ${\mathcal {M}}$ -subobjects

Lawvere doctrines are a central concept in categorical logic, dating back to its origins [30, 31]. We recall here the relation to factorization systems, which is also standard as witnessed by [43, Exercise 4.3.5].

In an f-regular category $({\mathbf {E}}, {\mathcal {M}})$ , the arrows in ${\mathcal {M}}$ with codomain X (better, their equivalence classes under isomorphisms) are called the ${\mathcal {M}}$ -subobjects of X and we denote them as $S\buildrel s\over \hookrightarrow X$ . We think of S as a ‘subspace’ of X. The ${\mathcal {M}}$ -subobjects of X are preordered in the standard way: Given $s_1 : S_1 \hookrightarrow X$ and $s_2 : S_2 \hookrightarrow X$ , we write $S_1 \leq S_2$ if there is $h : S_1 \longrightarrow S_2$ such that $hs_2 = s_1$ . Since $s_2$ is monomorphic, such an h is unique, and it is also in ${\mathcal {M}}$ by Lemma 2.4 below.

Lemma 2.4. In an f-regular category $({\mathbf {E}}, \mathcal {M})$ , the first component of a map in $ \mathcal {M}$ is in $ \mathcal {M}$ too.

Proof. Let $l\circ h\in {\mathcal {M}}$ and write h as $m\circ e$ with $e \in {\mathcal {M}}^\bot $ and $m \in {\mathcal {M}}$ . By orthogonality of e and $l\circ h$ , we obtain an arrow f making the following diagram commute:

This means that e is a split monomorphism, hence a regular monomorphism, hence in ${\mathcal {M}}$ . Since e is also in ${\mathcal {M}}^\bot $ , it is an isomorphism. Thus $h = m\circ e \in {\mathcal {M}}$ .

If $S_1 \leq S_2$ and $S_2 \leq S_1$ , then $S_1$ and $S_2$ differ by an isomorphism and we say that they are equivalent as ${\mathcal {M}}$ -subobjects: we denote by $\operatorname {\mathrm {Sub}}_{\mathcal {M}}(X)$ the equivalence classes of ${\mathcal {M}}$ -subobjects of X. By pullback stability, taking pullbacks yields a contravariant functor to the category of meet-semilattices. We indicate with $f^* : \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y) \longrightarrow \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X)$ the operation of pulling back along $f : X\longrightarrow Y$ . More is true indeed: we get a regular Lawvere doctrine structure, as defined below.

Definition 2.5. Let

be a commutative square in the category of posets. Suppose that $g^*$ and $u^*$ have left adjoints $\exists _g$ and $\exists _u$ . We say that the square is Beck–Chevalley if $f^*\circ \exists _g = \exists _u\circ v^*$ .

Below, we let $\mathbf {SMLat}$ be the category of semilattices and related morphisms.

Definition 2.6. A regular Lawvere doctrine is a pair $({\mathbf {E}}, \mathcal {P})$ given by a lex category ${\mathbf {E}}$ and a functor $\mathcal {P} : {\mathbf {E}}^{\mathrm {op}}\longrightarrow \mathbf {SMLat}$ such that:

• • For every $f : X\longrightarrow Y$ in ${\mathbf {E}}$ , the function $f^* = \mathcal {P}(f) : \mathcal {P}(Y)\longrightarrow \mathcal {P}(X)$ has a left adjoint $\exists _f : \mathcal {P}(X) \longrightarrow \mathcal {P}(Y)$ .

• • For every $f : X\longrightarrow Y$ in ${\mathbf {E}}$ , the so-called Frobenius condition holds:

  $$\begin{align*}\exists_f(\varphi \land f^*(\psi)) = (\exists_f \varphi) \land \psi \text{.} \end{align*}$$

• • Every pullback square is sent to a Beck–Chevalley square.

Proposition 2.7. If $({\mathbf {E}}, {\mathcal {M}})$ is an f-regular category, then $\operatorname {\mathrm {Sub}}_{\mathcal {M}}$ is a regular Lawvere doctrine.

Proof. Let $f : X\longrightarrow Y$ be an arrow in ${\mathbf {E}}$ . We first show that $f^* : \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y) \longrightarrow \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X)$ has a left adjoint $\exists _f : \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X) \longrightarrow \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y)$ . Let $s : S \hookrightarrow X$ in ${\mathcal {M}}$ . Then $\exists _f S \in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y)$ is obtained by factoring $sf$ as an arrow $e : S \longrightarrow \exists _f S$ in ${\mathcal {M}}^\bot $ followed by an arrow $m : \exists _f S \hookrightarrow Y$ in ${\mathcal {M}}$ . Let $T \hookrightarrow Y$ in ${\mathcal {M}}$ . We want to show that in the diagram below, the dashed arrow u exists if and only if the dashed arrow v exists.

If v exists, then u exists too by the universal property of the pullback defining $f^* T$ . On the other hand, if u exists, then v exists too by applying the orthogonality of $S \longrightarrow \exists _f S$ and $T \hookrightarrow Y$ in the diagram below.

In order to check the Beck–Chevalley condition, consider a commutative square as below in ${\mathbf {E}}$ .

Given $s : S \hookrightarrow Y_1$ in ${\mathcal {M}}$ , take the further pullback below.

We have to show that the $({\mathcal {E}}, {\mathcal {M}})$ -factorization of $s' p_2$ is just (up to an isomorphism) the factorization of $s f_1$ pulled back along $f_2$ . But this is just the stability of the factorization applied to the pullback square below, obtained by pasting the two squares above.

As for the Frobenius condition, it is actually a special case of the Beck–Chevalley condition. We recall here the well-known argument. Let $f : X\longrightarrow Y$ in ${\mathbf {E}}$ , let $S \in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y),$ and let $T \in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X)$ . We consider the pullback square below.

Applying the Beck–Chevalley condition to $T \in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X)$ , we obtain the equation in $\operatorname {\mathrm {Sub}}_{\mathcal {M}}(S)$

$$\begin{align*}\exists_g j^* T = i^* \exists_f T \text{.} \end{align*}$$

Notice that $j^*$ and $i^*$ are given by intersecting with $f^* S$ and S, respectively. Hence we obtain

$$\begin{align*}\exists_g(T \land f^* S) = (\exists_f T) \land S \text{.}\end{align*}$$

Composing both of these subobjects of S with $S \hookrightarrow Y$ , and noticing that the left-hand side becomes the image of $T \land f^* S \longrightarrow Y$ , we obtain the desired equality

$$\begin{align*}\exists_f (T \land f^*S) = (\exists_f T) \land S \text{.}\\[-31pt] \end{align*}$$

In fact, as shown in Proposition 2.8 below, f-regular categories form in this way a (non-full) sub-category of regular Lawvere doctrines. Given two regular Lawvere doctrines $\mathcal {P}_1 : {\mathbf {E}}_1^{\mathrm {op}}\longrightarrow \mathbf {SMLat}$ and $\mathcal {P}_2 : {\mathbf {E}}_2^{\mathrm {op}}\longrightarrow \mathbf {SMLat}$ , a morphism $\mathcal {P}_1 \longrightarrow \mathcal {P}_2$ is a pair $(\Phi , \alpha )$ given by lex functor $\Phi : {\mathbf {E}}_1 \longrightarrow {\mathbf {E}}_2$ and a natural transformation ${\alpha : \mathcal {P}_1 \longrightarrow \mathcal {P}_2 \circ \Phi }$ such that all the naturality squares below (induced by some arrow $X\longrightarrow Y$ in ${\mathbf {E}}_1$ ) are Beck–Chevalley.

Proposition 2.8. Let $({\mathbf {E}}_1,{\mathcal {M}}_1)$ and $({\mathbf {E}}_2,{\mathcal {M}}_2)$ be two f-regular categories. If ${\Phi : {\mathbf {E}}_1 \longrightarrow {\mathbf {E}}_2}$ is f-regular, then there is a unique natural transformation ${\alpha : \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1} \longrightarrow \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_2} \circ \Phi }$ such that $(\Phi ,\alpha )$ is a morphism of Lawvere doctrines.

Proof. Let $\Phi : {\mathbf {E}}_1 \longrightarrow {\mathbf {E}}_2$ be an f-regular functor. We first show that there is at most one $\alpha : \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1} \longrightarrow \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_2} \circ \Phi $ such that $(\Phi ,\alpha )$ is a morphism of Lawvere doctrines. Let $S \in \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1}(X)$ . Consider the following naturality square of a natural transformation $\alpha : \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1} \longrightarrow \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_2} \circ \Phi $ .

Recall that in this article, meet-semilattices have a top element and homomorphisms preserve it. The Beck–Chevalley condition applied to $1_S \in \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1}(S)$ says that $\alpha _X(S) = \Phi (S)$ (i.e., $\alpha _X$ maps $S\hookrightarrow X$ to the subobject $\Phi (S) \hookrightarrow \Phi (X)$ ). Hence $\alpha $ is uniquely determined by $\Phi $ and $\alpha _X(S) = \Phi (S)$ .

Next, we check that putting $\alpha _X(S) = \Phi (S)$ always defines a natural transformation whose naturality squares are Beck–Chevalley. First, $\Phi (S) \in \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_2}(\Phi (X))$ whenever $S \in \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1}(X)$ since $\Phi $ preserves ${\mathcal {M}}$ . Since $\Phi $ is left exact, $\alpha _X$ preserves meets. Moreover, $\Phi $ preserves pullbacks, hence $\Phi (f)^* \Phi (S) = \Phi (f^* S)$ and $\alpha $ is a natural transformation. In order to check that the naturality squares are Beck–Chevalley, let $f : X\longrightarrow Y$ in ${\mathbf {E}}_1$ and let $S \in \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1}(X)$ . Since $\Phi $ preserves the factorization of arrows, we obtain that $\Phi (\exists _f S) = \exists _{\Phi (f)} \Phi (S)$ . This concludes the proof that $(\Phi ,\alpha )$ is a morphism of Lawvere doctrines.

We now show a converse of Proposition 2.8 when $({\mathbf {E}}_2,{\mathcal {M}}_2)$ comes from a regular category such as $\mathbf {Set}$ .

Proposition 2.9. Let $({\mathbf {E}}_1,{\mathcal {M}}_1)$ and $({\mathbf {E}}_2,{\mathcal {M}}_2)$ be two f-regular categories and suppose that ${\mathcal {M}}_2$ is exactly the class of monomorphisms of ${\mathbf {E}}_2$ . Then every Lawvere doctrine morphism $\operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1} \longrightarrow \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_2}$ is induced by an f-regular functor ${\mathbf {E}}_1 \longrightarrow {\mathbf {E}}_2$ as in Proposition 2.8.

Proof. Let $\Phi : {\mathbf {E}}_1 \longrightarrow {\mathbf {E}}_2$ be a lex functor and let $\alpha : \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_1} \longrightarrow \operatorname {\mathrm {Sub}}_{{\mathcal {M}}_2} \circ \Phi $ be a natural transformation whose naturality squares are Beck–Chevalley. We want to show that $\Phi $ respects the factorization systems. Any $f \in {\mathcal {M}}_1$ is a monomorphism, hence $\Phi (f)$ is also a monomorphism and it is in ${\mathcal {M}}_2$ . Suppose now that $f : X\longrightarrow Y$ is in the left class of ${\mathbf {E}}_1$ . This means that $\exists _f X = Y$ . Using that $\alpha $ preserves the top elements and applying the Beck–Chevalley condition on the naturality square associated with f, we obtain that $\exists _{\Phi (f)} \Phi (X) = \Phi (Y)$ , which means that $\Phi (f)$ is in the left class of ${\mathbf {E}}_2$ .

Remark 2.10. The stable orthogonal factorization systems $({\mathcal {E}}, {\mathcal {M}})$ on a lex category ${\mathbf {E}}$ such that ${\mathcal {M}}$ is included in the class of monomorphisms correspond to the regular Lawvere doctrines with full comprehension as defined in [33, 40]. The class of regular monomorphisms is included in ${\mathcal {M}}$ if and only if every morphism of ${\mathbf {E}}$ is determined uniquely by its graph in the internal language.

Coherent logic is obtained from regular logic by adding finite joins. For instance, a coherent Lawvere doctrine is a regular doctrine valued in the category of distributive lattices. We define similarly f-coherent categories.

Definition 2.11. An f-coherent category is an f-regular category $({\mathbf {E}}, {\mathcal {M}})$ whose functor $\operatorname {\mathrm {Sub}}_{{\mathcal {M}}}$ is valued in the category of distributive lattices. An f-regular functor between f-coherent categories is f-coherent when it moreover preserves finite joins of ${\mathcal {M}}$ -subobjects. We say that $({\mathbf {E}}, {\mathcal {M}})$ is f-Boolean when $\operatorname {\mathrm {Sub}}_{\mathcal {M}}$ is valued in the category of Boolean algebras.

3 Adding modalities

A modal lattice is a distributive lattice $(D, \wedge , \top , \vee , \bot )$ endowed with an operator $\Diamond : D\longrightarrow D$ s.t. the equalities

$$ \begin{align*}\Diamond(x\vee y)=\Diamond x \vee \Diamond y,\qquad \Diamond \bot=\bot \end{align*} $$

hold for all $x,y\in D$ . The typical example of a modal lattice (which is also a Boolean algebra) comes from a Kripke frame, which is a pair $(W,R)$ given by a set endowed with a binary relation $R\subseteq W\times W$ . Given such a Kripke frame, we can define a diamond operator:

$$ \begin{align*}\Diamond_R: \wp(W) \longrightarrow \wp(W) \end{align*} $$

by taking

(3) $$ \begin{align} \Diamond_R (S)~:=~\{w\in W\mid {\text{there is}}~ v\in S ~\text{s.t.}~ wRv\}. \end{align} $$

Notice that the definition of $\Diamond _R$ makes sense also whenever the domain and the codomain of the relation R do not coincide: in such a case, if $R\subseteq W'\times W$ , then the operator

(4) $$ \begin{align} \Diamond_R: \wp(W) \longrightarrow \wp(W') \end{align} $$

is a hemimorphism, i.e., it preserves finite joins. In the following, we shall make use of operators like (4) defined via (3) for relations $R\subseteq W'\times W$ .

This is our main definition.

Definition 3.1. A modal category is an f-coherent category $({\mathbf {E}}, {\mathcal {M}})$ such that for every object X, the lattice $\operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X)$ is a modal lattice. Moreover, the following conditions must be satisfied:

• • (Continuity) for every $f:X\longrightarrow Y$ and $S\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(Y)$ we have

  (5) $$ \begin{align} \Diamond f^* S \leq f^* \Diamond S; \end{align} $$

• • (Subspace) for every $m:A\hookrightarrow X$ in $ \mathcal {M}$ and for every $S\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(A)$ it holds that

  (6) $$ \begin{align} \Diamond S=m^* \Diamond\, \exists_{m} \,S. \end{align} $$

The two axioms above are inspired by the definition of a continuous function and of a subspace in $\mathbf {Top}$ . The ‘continuity’ axiom (5) reflects the definition of a continuous function by means of the closure operator [28, Chapter 3, Theorem 1]. It can also be formulated as $\exists _f \Diamond S \leq \Diamond \exists _f S$ , for instance. The ‘subspace regularity’ axiom (6) reflects the definition of the induced topology on a subspace.

In all the examples below, the underlying factorization system is the epi/regular-mono factorization system.

Example 1. As expected, $\mathbf {Top}$ is a modal category: to see that subspaces are regular monos one may use for instance the two element set with the co-discrete topology as a regular subobject classifier. As a modal category, $\mathbf {Top}$ satisfies the following additional conditions:

• • (Closure) for every X and $S \in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X)$

  (7) $$ \begin{align} S \leq \Diamond S,\qquad \Diamond \Diamond S \leq \Diamond S; \end{align} $$

• • (Variable Independence) for every $X_1$ , $X_2$ and $S_1\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X_1)$ , $S_2\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X_2)$ , we have

  (8) $$ \begin{align} (\pi_{X_1}^* \Diamond S_1) \wedge (\pi_{X_2}^* \Diamond S_2) = \Diamond (\pi_{X_1}^* S_1 \wedge \pi_{X_2}^* S_2). \end{align} $$

These conditions are due to specific properties of the closure operator and of the product topology (variable independence implies in particular that the projection maps $\pi _{X_1}, \pi _{X_2}$ are open in the sense that $\pi _{X_1}^*, \pi _{X_2}^*$ fully preserve the $\Diamond $ closure operator). We shall turn to the question of whether the above conditions are sufficient for an embedding theorem into $\mathbf {Top}$ in another paper [19] (a partial positive result, for function-free first-order one-sorted languages has been obtained in [16, 20]).

Example 2. Recall that a Kripke frame, as defined above, is a pair $(W,R)$ with W a set and $R \subseteq W\times W$ a binary relation. Given two Kripke frames $(W,R)$ and $(W',R')$ , a map $f : W \longrightarrow W'$ is called stable if $x R y \implies f(x) R' f(y)$ (the definition is taken from [4, Definition 3.2], forgetting the topology). Then the category of Kripke frames and stable maps is a modal category, where ${\mathcal {M}}$ consists of the embeddings, i.e., the injective stable maps f such that $f(x) R' f(y) \implies x R y$ . Similarly, the category $\mathbf {POr}$ of preordered sets (= sets endowed with a reflexive and transitive relation) and order preserving maps is also a modal category.

Example 3. In this article, we concentrate on relational semantic environments similar to those introduced in [16, 20, 21] (see Section 1 for additional references). Given a graph

$$ \begin{align*}\mathbf{G} := \{d,c: A \rightrightarrows O\}\text{,} \end{align*} $$

a relational $\mathbf {G}$ -set X consists of the following data:

• • for every $\alpha \in O$ , a set $X_{\alpha }$ ;

• • for every $k\in A$ with $d(k)=\alpha $ and $ c(k)=\beta $ , a relation $X_k\subseteq X_{\alpha }\times X_{\beta }$ .

Below, we write to mean that $d(k)=\alpha , c(k)=\beta $ and $a\,k\,b$ to mean that $(a,b) \in X_k$ . A morphism of relational $\mathbf {G}$ -sets from X to Y is a family of functions

$$ \begin{align*}f:= ~\{f_{\alpha}: X_{\alpha}\longrightarrow Y_{\alpha}\}_{\alpha \in O} \end{align*} $$

that preserve the relations, i.e., such that

$$ \begin{align*}a\,k\,b \Rightarrow f_{\alpha}(a)\; k\; f_{\beta}(b)\end{align*} $$

for all , $a\in X_\alpha , b\in X_\beta $ . We denote by $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ the category of relational $\mathbf {G}$ -sets, with composition and the identities defined pointwise. This category has limits, also computed pointwise. The regular subobjects of a relational $\mathbf {G}$ -set X are the families

$$ \begin{align*}f:= ~\{S_{\alpha}\subseteq X_{\alpha}\}_{\alpha \in O} \end{align*} $$

with the relations $S_k$ obtained by taking the restrictions of the relations $X_k$ (they are classified by the two-element constant relational $\mathbf {G}$ -set with total relations). Epimorphisms are pointwise surjective functions, so that ${\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}}$ is an f-regular (actually an f-Boolean) category. ${\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}}$ has a natural structure of modal category: for S a regular subobject of X and $a\in X_{\alpha }$ we put

(9) $$ \begin{align} a\in (\Diamond S)_{\alpha} ~~\mathrm{iff}~~\mathrm{there~are~}k, b~ \text{ s.t. } akb ~\mathrm{and}~b\in S_{c(k)}. \end{align} $$

The next proposition shows the peculiarity of this example.

Proposition 3.2. ${\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}}$ is a quasi-topos.

Proof. In fact ${\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}}$ can be presented as the category of separated presheaves on a site $(\mathbf {C}, J)$ as follows (that separated presheaves are a quasi-topos is proved in [27, Section A2.6]). There are two kinds of objects in $\mathbf {C}$ : for each vertex $\alpha \in \mathbf {G}$ , there is an object $[\alpha ]$ in $\mathbf {C}$ ; for each edge in $\mathbf {G}$ , there is an object $[k]$ in $\mathbf {C}$ . For each edge in $\mathbf {G}$ , there are two arrows $[\alpha ]\longrightarrow [k]$ and $[\beta ]\longrightarrow [k]$ in $\mathbf {C}$ . These are the only non-identity arrows. The topology J consists of all the identity coverings, plus the coverings of the form $[\alpha ]\longrightarrow [k]$ and $[\beta ]\longrightarrow [k]$ for each edge in $\mathbf {G}$ . These coverings satisfy the axioms for a Grothendieck topology (see [27, Definition 2.1.9]).

A presheaf $X : \mathbf {C}^{\mathrm {op}} \longrightarrow \mathbf {Set}$ is separated iff $X([k])\subseteq X([\alpha ])\times X([\beta ])$ for each edge in $\mathbf {G}$ . This means that X is given by a family of sets $X([\alpha ])$ , plus a relation $X([k])$ for every edge k in $\mathbf {G}$ .

Example 4. Given a category $\mathbf {C}$ , we can consider relational presheaves [15, 38]: these are correspondences X associating with every object $\alpha $ of $\mathbf {C}$ a set $X_{\alpha }$ and with every arrow $k:\alpha \longrightarrow \beta $ in $\mathbf {C}$ a relation $X_k\subseteq X_{\alpha }\times X_{\beta }$ in such a way that the lax functoriality conditions

$$ \begin{align*}id_{\alpha} \subseteq X_{id_\alpha} \qquad X_k\circ X_l \subseteq X_{l\circ k} \end{align*} $$

hold, for every $\alpha $ and every pair of composable arrows $k,l$ . The arrows of the category $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {C}}$ of relational presheaves are defined in the same way as for ${\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}}$ . It turns out that $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {C}}$ is a modal category, but it is not a quasi-topos anymore: quasi-toposes are regular categories [27] and for $\mathbf {C}$ equal to the singleton category $\mathbf {1}$ we have $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {1}}\simeq \mathbf {POr}$ which is not regular.

Example 5. There are exactly three (all uninteresting!) structures of modal category on $\mathbf {Set}$ with its standard factorization system of surjections and injections: either (a) we have $\Diamond S = S$ for all S, or (b) we have $\Diamond S = \bot $ for all S, or (c) we have $\Diamond S=\top $ for all nonempty S. In order to show this, we first consider the case where there are X and $S\subseteq X$ such that $\Diamond S \not \subseteq S$ . In such case, there is also $x \in X$ , $x\in \Diamond S$ , $x \notin S$ . We show that then we are in case (c), i.e., we have $\Diamond T = \top $ for all Y and all nonempty $T \subseteq Y$ : to see this, take $t \in T$ and pick any $y \in Y$ . Let $f : X\longrightarrow Y$ the function defined by $\exists _f(S)=\{t\}$ and $\exists _f(X\setminus S) = \{y\}$ . Applying the continuity axiom to f, we get $y \in \exists _f(\Diamond S) \subseteq \Diamond \exists _f(S)= \Diamond \{t\}$ , thus $\top =\Diamond \{t\}\subseteq \Diamond T$ because y is arbitrary. We are left with the case where $\Diamond S \subseteq S$ holds for all X and all $S \subseteq X$ . If $\Diamond S$ is always empty, we are done because we are in case (b). To conclude, we must show that we are in case (a), namely that if there is $x \in \Diamond S$ for some $S \subseteq X$ , then we have also $\Diamond \{y\} = \{y\}$ for all Y and $y \in Y$ : Let $f : X\longrightarrow Y$ be the constant map at y. By continuity we get $y \in \exists _f(\Diamond S) \subseteq \Diamond \exists _f(S) \subseteq \Diamond \{y\}$ . This gives $\Diamond \{y\} \supseteq \{y\}$ and also $\Diamond \{y\} = \{y\}$ , because $\Diamond \{y\} \subseteq \{y\}$ comes from the fact that $\Diamond S \subseteq S$ holds for all S.

4 The representation theorem

Recall that a morphism $\Phi $ between f-coherent categories $\Phi :({{\mathbf {E}}_1}, \mathcal {M}_1)\longrightarrow ({{\mathbf {E}}_2}, \mathcal {M}_2)$ is a lex functor that preserves factorizations and joins; if $({{\mathbf {E}}_1}, \mathcal {M}_1)$ and $({{\mathbf {E}}_2}, \mathcal {M}_2)$ are modal categories, a modal morphism between them is an f-coherent functor $ \Phi : ({{\mathbf {E}}_1}, \mathcal {M}_1)\longrightarrow ({{\mathbf {E}}_2}, \mathcal {M}_2)$ that also preserves the modal operators.

We say that $\Phi $ is a conservative (resp., $ \mathcal {M}$ -conservative) embedding when for any monomorphism m (resp., $m\in \mathcal {M}_1$ ), if $\Phi (m)$ is iso, then m is iso as well. For instance, the forgetful functor from $\mathbf {Top}$ to $\mathbf {Set}$ is an $ \mathcal {M}$ -conservative embedding, but not a conservative embedding. We shall see that it is always possible to embed an f-regular category $({\mathbf {E}}, \mathcal {M})$ into a power of $\mathbf {Set}$ in an $ \mathcal {M}$ -conservative way, but not necessarily in a conservative way, because monos which are also in ${\mathcal {M}}^\bot $ become automatically isomorphisms in a topos like $\mathbf {Set}$ and its powers.

We shall prove a representation theorem of modal categories into categories of relational $\mathbf {G}$ -sets: the construction will be very similar to the celebrated Joyal construction for the completeness theorem of intuitionistic first-order logic [34, p. 75]. The representation will take the form of an $ \mathcal {M}$ -conservative embedding, to be strengthened to a conservative embedding, in the case of modal categories satisfying the saturation condition (see Section 5 below). In order to prove such a theorem, we first fix some notation about models in $\mathbf {Set}$ and review some basic material concerning Lawvere doctrines.

A coherent model (or just a model) of an f-coherent category $({\mathbf {E}}, \mathcal {M})$ is an f-coherent functor to $\mathbf {Set}$ equipped with its usual factorization system of surjections and injections. The class of the models of $({\mathbf {E}}, \mathcal {M})$ is denoted by $\operatorname {\mathrm {\mathbf {Mod}}}({\mathbf {E}}, \mathcal {M})$ . Thanks to Proposition 2.9 which transfers to f-coherent categories and coherent Lawvere doctrines, these models coincide with the models of the associated coherent Lawvere doctrine.

To make notation easier, for a given model $M : {\mathbf {E}} \longrightarrow \mathbf {Set}$ , we shall often use $X_M, f_M, {\ldots }$ instead of $M(X), M(f),{\ldots }$ to indicate the values of the functor M on objects and arrows of ${\mathbf {E}}$ . For a subobject $s : S \hookrightarrow X$ in $\operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X)$ , we might also use the notation $\lbrack {\kern-1.3pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack _M$ for the subset of $M(X)$ given by image of the function $M(s) : M(S) \hookrightarrow M(X)$ (this is the subset of $M(X)$ canonically representing the subobject $M(s) : M(S) \hookrightarrow M(X)$ ). Equalities like $\lbrack {\kern-1pt}\lbrack \exists _f S\rbrack {\kern-1.3pt}\rbrack _M= \exists _{f_M}\lbrack {\kern-1pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack _M$ and $\lbrack {\kern-1pt}\lbrack f^* T\rbrack {\kern-1.3pt}\rbrack _M= f_M^*\lbrack {\kern-1pt}\lbrack T\rbrack {\kern-1.3pt}\rbrack _M$ are immediate consequences of our definitions and notational conventions (they will be frequently used in this section).

We need a model existence theorem for Lawvere doctrines, to be applied to doctrines of the kind $({\mathbf {E}}, \operatorname {\mathrm {Sub}}_{ \mathcal {M}})$ for an f-coherent category $({\mathbf {E}}, \mathcal {M})$ . Recall that a filter in a distributive lattice $(D, \wedge , \top , \vee , \bot )$ is a subset $F\subseteq D$ that contains $\top $ , is closed under $\wedge $ and is upward-closed relatively to the ordering $\leq $ of the lattice. Dually, an ideal is a downward-closed subset containing $\bot $ and closed under $\vee $ . A filter F is prime if it does not contain $\bot $ and contains either x or y in case it contains $x\vee y$ . The existence of enough prime filters is a consequence of the following well-known lemma (whose validity depends on the axiom of choice).

Lemma 4.1. If $\Gamma $ and $\Delta $ be two disjoint subsets of a distributive lattice D such that $\Gamma $ is a filter and $\Delta $ is an ideal, then there exists a prime filter P such that $P\supseteq \Gamma $ and $P\cap \Delta =\emptyset $ .

Let $({\mathbf {E}}, \mathcal {P})$ be a coherent Lawvere doctrine and let X be an object of ${\mathbf {E}}$ ; a prime filter P of $\mathcal {P}(X)$ is called an X-type of $\mathcal {P}$ . Typical X-types come from models: if $(M,\alpha )$ is a model and $x\in M(X)$ , the set P of the $S\in \mathcal {P}(X)$ such that $x\in \alpha _X(S)$ is a type; we say in this case that x realizes P. All types can in fact be realized, as stated in the following lemma, which is nothing but a small variant of the Gödel completeness theorem for first-order logic (see, e.g., [23, Theorem 7.2]).

Proposition 4.2. Given a a coherent Lawvere doctrine $({\mathbf {E}}, \mathcal {P})$ and an X-type P of $\mathcal {P}$ , there is a model $(M,\alpha )$ of $({\mathbf {E}}, \mathcal {P})$ and an element $x\in M(X)$ realizing P.

Proposition 4.2 can be used to prove a representation theorem for small f-coherent categories into powers of $\mathbf {Set}$ as follows. Notice that given any f-coherent functor $\Phi : ({\mathbf {E}}, \mathcal {M}) \longrightarrow \mathbf {Set}^W$ from $({\mathbf {E}}, \mathcal {M})$ into a W-power of $\mathbf {Set}$ and given $w\in W$ , the composite functor

$$ \begin{align*}ev_w\circ \Phi: ({\mathbf{E}}, \mathcal{M}) \longrightarrow \mathbf{Set}^W \longrightarrow \mathbf{Set} \end{align*} $$

is an f-coherent model of $({\mathbf {E}}, \mathcal {M})$ (here $ev_w$ is the “evaluation at w” functor, i.e., the functor induced by the inclusion $\{w\}\hookrightarrow W$ ). If we want to show that $({\mathbf {E}}, \mathcal {M})$ has an $ \mathcal {M}$ -conservative embedding into a power of $\mathbf {Set}$ , the obvious idea is to take W to be the class $\operatorname {\mathrm {\mathbf {Mod}}}({\mathbf {E}}, \mathcal {M})$ of all the f-coherent models of $({\mathbf {E}}, \mathcal {M})$ . ² We are lead to the following,

Theorem 4.3. For every small f-coherent category $({\mathbf {E}}, \mathcal {M})$ , the evaluation functor

$$ \begin{align*}ev_{({\mathbf{E}}, \mathcal{M})}= (ev_M \mid M\in \operatorname{\mathrm{\mathbf{Mod}}}({\mathbf{E}}, \mathcal{M})) : {\mathbf{E}} \longrightarrow \mathbf{Set}^{\operatorname{\mathrm{\mathbf{Mod}}}({\mathbf{E}}, \mathcal{M})} \end{align*} $$

is an $ \mathcal {M}$ -conservative embedding of f-coherent categories.

Proof. Trivially, limits and factorizations are all preserved because they are computed pointwise in powers of $\mathbf {Set}$ . $ \mathcal {M}$ -conservativity (but not conservativity!) is ensured by Proposition 4.2: if $S\buildrel {s}\over \hookrightarrow X$ is in $ \mathcal {M}$ but is not an iso, then $S\neq \top $ in $\operatorname {\mathrm {Sub}}_{ \mathcal {M} }(X)$ , hence there is an X-type P such that $S\not \in P$ by Lemma 4.1. If P is realized in M by some $a\in M(X)$ , we have that $a\not \in \lbrack {\kern-1pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack _M$ , which means that the image of $M(s)$ is not $M(X)$ . Thus $ev_{({\mathbf {E}}, \mathcal {M})}(s)$ is not iso because its M-component is not such.

In order to extend the above machinery to modal categories, we need a notion of a modal transformation between two f-coherent models of a modal category. This notion will be the modal analogue of a natural transformation—aka elementary embedding—among two models of a classical first-order theory. We again take inspiration from modal morphisms of the kind $\Phi :({\mathbf {E}}, \mathcal {M})\longrightarrow {\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf { G}}}$ . Notice that for every in $\mathbf {G}$ such a modal morphism $\Phi $ produces a composite functor

by composition with the functor induced by the sub-graph inclusion . This composite functor represents k as a family of relations (indexed by the objects of ${\mathbf {E}}$ ) between two coherent models of $({\mathbf {E}}, {\mathcal {M}})$ . Such a family of relations satisfies the following definition, thus motivating it.

Formally, we define a modal transformation $R: M\longrightarrow N$ between two coherent models

$$ \begin{align*} M: (\mathrm{E}, \mathcal{M}) \longrightarrow \mathbf{Set} \text{ and } N: ({\mathrm{E}}, \mathcal{M}) \longrightarrow \mathbf{Set} \end{align*} $$

of a modal category $({\mathbf {E}}, \mathcal {M})$ as a lex subfunctor (i.e., a subfunctor preseving finite products and equalizer)

$$ \begin{align*}R\subseteq M\times N \end{align*} $$

satisfying the ‘reflection’ condition

(10) $$ \begin{align} s_M(a) \; R(X) \; s_N(b) ~~\Rightarrow~~ a \; R(S) \; b \end{align} $$

(for all $S\buildrel {s}\over \hookrightarrow X\in \mathcal {M}$ , $a\in S_M, b\in S_N$ ) and the ‘continuity’ condition:

(11) $$ \begin{align} \Diamond_{R(X)} \lbrack {\kern-1.3pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack_N \;\subseteq\; \lbrack {\kern-1.3pt}\lbrack \Diamond S\rbrack {\kern-1.3pt}\rbrack_M. \end{align} $$

(for all X and $S\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X)$ ).

Remark 4.4. Unraveling condition (11) according to (3) and (4), we can reformulate it as

(12) $$ \begin{align} a \,R(X)\,b \mathrm{and}~b\in \lbrack {\kern-1.3pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack_N \Rightarrow a\in \lbrack {\kern-1.3pt}\lbrack \Diamond S\rbrack {\kern-1.3pt}\rbrack_M \end{align} $$

(for all $a\in X_M$ and $b\in X_N$ ). This is clearly reminiscent of the definition of the accessibility relation in canonical models widely used in the modal logic literature [9].

Remark 4.5. We defined a modal transformation to be a lex subfunctor satisfying reflection and continuity conditions. This definition is slightly redundant, because reflection implies preservation of equalizers (recall that equalizers are in $ \mathcal {M}$ ). On the other hand, if $ \mathcal {M}$ is the class of regular monos (as it often happens in the semantic examples), then reflection becomes redundant.

We let $\mathbf {G}({\mathbf {E}}, \mathcal {M})$ be the (large) graph of all the coherent models of a modal category $({\mathbf {E}}, \mathcal {M})$ and of all the modal transformations among them. We have an evaluation functor

$$ \begin{align*}ev_{({\mathbf{E}}, {\mathcal{M}})}: {\mathbf{E}} \longrightarrow \mathbf{Rel^{G(E, {\mathcal{M}})}} \end{align*} $$

mapping an object X in ${\mathbf {E}}$ to the relational $\mathbf {G}({\mathbf {E}}, \mathcal {M})$ -set $ev_{({\mathbf {E}}, \mathcal {M})}(X)$ whose M-component is $X_M$ and whose R-component (for in $\mathbf { G}({\mathbf {E}}, \mathcal {M})$ ) is $R(X)$ . The action of the functor $ev_{({\mathbf {E}}, \mathcal {M})}$ on an arrow $f:X\longrightarrow Y$ is defined by $ev_{({\mathbf {E}}, \mathcal {M})}(f)(M) = f_M$ . Our representation theorem now reads as follows.

Theorem 4.6. For every small modal category $({\mathbf {E}}, \mathcal {M})$ , the evaluation functor

$$ \begin{align*}ev_{({\mathbf{E}}, {\mathcal{M}})}: ({\mathbf{E}}, {\mathcal{M}}) \longrightarrow \mathbf{Rel^{G(E, \mathcal{M})}} \end{align*} $$

is an $ \mathcal {M}$ -conservative embedding of modal categories.

The proof of Theorem 4.6 occupies the remaining part of this section. We fix a small modal category $({\mathbf {E}}, \mathcal {M})$ . We spell out in detail the conditions for R to be a modal transformation between two models M and N of ${\mathbf {E}}$ . Below we often omit types for simplicity and write simply $a R b$ when $a\in X_M, b\in Y_N$ and $(a,b) \in R(X)$ ; similarly, we write just $f(a)$ for $f_M(a)$ , when $f:X\longrightarrow Y$ is an arrow in ${\mathbf {E}}$ and $a\in X_M$ . First of all, R is in fact a family of relations $R=\{ R(X)\subseteq X_M\times X_N\}_X$ indexed by the objects of ${\mathbf {E}}$ ; the fact that such a family R is a lex subfunctor and satisfies reflection means that the following conditions are satisfied:

(13) $$ \begin{align} & a_1Rb_1 \;\& \cdots \&\; a_nRb_n \Rightarrow (a_1, \dots, a_n) R (b_1, \dots, b_n) ~(\mathrm{for}~n\geq 0) \end{align} $$

(14) $$ \begin{align} & a Rb \Rightarrow f(a) R f(b) ~~(\mathrm{for~all}~f~{\text{of appropriate domain in}}~{\mathbf{E}}) \end{align} $$

(15) $$ \begin{align} & s(a) R s(b) \Rightarrow a Rb,~~(\mathrm{for}~ S\buildrel{s}\over \hookrightarrow X \in \operatorname{\mathrm{Sub}}_{ \mathcal{M}}(X)). \end{align} $$

In addition we have condition (11), namely

(16) $$ \begin{align} aRb \text{ and } b\in \lbrack {\kern-1.3pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack_N \Rightarrow a\in \lbrack {\kern-1.3pt}\lbrack \Diamond S\rbrack {\kern-1.3pt}\rbrack_M. \end{align} $$

We will sometimes write $X_R$ instead of $R(X)$ , using the same convention for modal transformations as for models.

An $ \mathcal {M}$ -partial map is a span where $S\hookrightarrow X$ is in $ \mathcal {M}$ .

Lemma 4.7. Given two models $M,N$ and a family of relations $R=\{ R(X)\subseteq M(X)\times N(X)\}_X$ indexed by the objects of ${\mathbf {E}}$ , the least family of relations $\tilde {R}$ containing R and satisfying conditions (14) and (15) can be described as follows.

For $a\in Y_M, b\in Y_N$ we have $a {\tilde {R}}b$ iff there exists a span

where $s \in {\mathcal {M}}$ and $a' \in S_M, b' \in S_N$ such that:

• • $f(a') = a$ and $f(b') = b$ ;

• • $s(a')~R~s(b')$ .

Proof. The proof is a simple check, using the stability of $ \mathcal {M}$ under pullbacks and the fact arrows in $ \mathcal {M}$ compose.

Lemma 4.8. If, for two given models $M,N$ , a family R of relations indexed by the objects of ${\mathbf {E}}$ satisfies condition (16), then so does the family of relations $\tilde {R}$ mentioned in Lemma 4.7.

Proof. This is due to the continuity and subspace conditions of Definition 3.1. Suppose in fact that we have $a {\tilde {R}}b$ so that there exists the data mentioned in the statement of Lemma 4.7. Suppose that $b\in \lbrack {\kern-1pt}\lbrack U\rbrack {\kern-1.3pt}\rbrack _N$ for some $U\buildrel {u}\over {\hookrightarrow } Y$ . From $f(b')=b$ we obtain

$$ \begin{align*}b' \in f_N^*\lbrack {\kern-1.3pt}\lbrack U\rbrack {\kern-1.3pt}\rbrack_N = \lbrack {\kern-1.3pt}\lbrack f^* U\rbrack {\kern-1.3pt}\rbrack_{N}; \end{align*} $$

thus

$$ \begin{align*}s(b') \in \exists_{s_N} \lbrack {\kern-1.3pt}\lbrack f^* U\rbrack {\kern-1.3pt}\rbrack_{N}= \lbrack {\kern-1.3pt}\lbrack \exists_{s} f^* U\rbrack {\kern-1.3pt}\rbrack_{N}. \end{align*} $$

Since R satisfies (16) and $s(a')~R~s(b')$ , we obtain

$$ \begin{align*}s(a') \in \lbrack {\kern-1.3pt}\lbrack \Diamond\exists_{s} f^* U\rbrack {\kern-1.3pt}\rbrack_{M} \end{align*} $$

and

$$ \begin{align*}a' \in {s}_M^* \lbrack {\kern-1.3pt}\lbrack \Diamond\exists_{s} f^* U \rbrack {\kern-1.3pt}\rbrack_{M}= \lbrack {\kern-1.3pt}\lbrack s^*\Diamond\exists_{s} f^* U \rbrack {\kern-1.3pt}\rbrack_{M}. \end{align*} $$

By subspace and continuity conditions from Definition 3.1, we have

$$ \begin{align*}a' \in \lbrack {\kern-1.3pt}\lbrack s^* \Diamond\exists_{s} f^* U\rbrack {\kern-1.3pt}\rbrack_{M}= \lbrack {\kern-1.3pt}\lbrack \Diamond f^* U\rbrack {\kern-1.3pt}\rbrack_{M} \subseteq \lbrack {\kern-1.3pt}\lbrack f^*\Diamond U\rbrack {\kern-1.3pt}\rbrack_{M}= f_M^* \lbrack {\kern-1.3pt}\lbrack \Diamond U\rbrack {\kern-1.3pt}\rbrack_{M}. \end{align*} $$

Now, $f(a')=a$ yields $a\in \lbrack {\kern-1pt}\lbrack \Diamond U\rbrack {\kern-1.3pt}\rbrack _{M}$ , as desired.

In the following lemma, we say that the family of relations R is a singleton if only one of the $R(X)$ is non-empty and that this one contains a single element.

Lemma 4.9. Let $M,N$ be two models of ${\mathbf {E}}$ and let R be a family of relations indexed by the objects of ${\mathbf {E}}$ . Suppose that either R satisfies (13), or that R is a singleton. Then the family $\tilde {R}$ defined in Lemma 4.7 satisfies (13).

Proof. Suppose that R satisfies (13). The only element of $\mathbf {1}_M \times \mathbf {1}_N$ (here $\mathbf {1}$ is the terminal object of ${\mathbf {E}}$ ) is in $\tilde {R}$ because it is in R. The fact that $\tilde {R}$ is closed under binary products is seen by observing that the product of two diagrams of the shape presented in Lemma 4.7 is again of the same type, because $s_1 \times s_2 \in {\mathcal {M}}$ whenever $s_1, s_2 \in {\mathcal {M}}$ .

Suppose that R is a singleton and contains only $(a,b) \in X_M \times X_N$ . Let $R_p$ be the closure of R under (13). We first show that $R_p \subseteq \tilde {R}$ . Indeed, for any $n\geq 0$ , the image of $(a,b)$ by the diagonal map $X_M \times X_N \longrightarrow X_M^n \times X_N^n$ is in $\tilde {R}$ , and $R_p$ consists precisely of all these elements. Since $R_p \subseteq \tilde {R}$ , we obtain that $\tilde {R_p} = \tilde {R}$ because $\tilde {R_p} \subseteq \tilde {\tilde {R}} = \tilde {R}$ . Hence $\tilde {R_p}$ satisfies (13) by the first part of this lemma.

We can now conclude the proof of Theorem 4.6.

Proof. We must prove that the evaluation functor is $ \mathcal {M}$ -conservative and preserves all the operations. $ \mathcal {M}$ -conservativity is proved in the same way as in Theorem 4.3.

$\boldsymbol {ev}_{({{\mathbf{E}}}, \mathcal {M})}$ preserves finite limits. For any coherent model M, the functor ${X \mapsto M(X)}$ preserves finite limits. For any modal transformation $R \subseteq M\times N$ , the functor ${X \mapsto R(X)}$ preserves finite limits. Since limits in $\mathbf {Rel^{G({\mathbf {E}}, \mathcal {M})}}$ are computed pointwise, this shows that $ev_{({\mathbf {E}}, \mathcal {M})}$ preserves finite limits.

$\boldsymbol {ev}_{({{\mathbf{E}}}, \mathcal {M})}$ preserves the left class of the factorization system. The left class of the factorization system of $\mathbf {Rel^{G({\mathbf {E}}, \mathcal {M})}}$ consists of all the morphisms whose M-component is surjective for each coherent model M. If $f : X\longrightarrow Y$ is a morphism in the left class of ${\mathbf {E}}$ , then $M(f) : M(X) \longrightarrow M(Y)$ is surjective for each coherent model M. We get that $ev_{({\mathbf {E}}, \mathcal {M})}(f)$ is in the left class of $\mathbf {Rel^{G({\mathbf {E}}, \mathcal {M})}}$ .

$\boldsymbol {ev}_{({{\mathbf{E}}}, \mathcal {M})}$ preserves the right class of the factorization system. The right class of the factorization system of $\mathbf {Rel^{G({\mathbf {E}}, \mathcal {M})}}$ consists of all the morphisms $g : V\longrightarrow W$ such that $g_M : V_M \longrightarrow W_M$ is injective for each coherent model M, and moreover such that for each modal transformation $R \subseteq M\times N$ between coherent models, ${V_R \subseteq V_M \times V_N}$ is the restriction of $W_R \subseteq W_M \times W_N$ via $g_M \times g_N$ . If $f:X\longrightarrow Y$ is a morphism in the right class $ \mathcal {M}$ of ${\mathbf {E}}$ , then $f_M:X_M \longrightarrow Y_M$ is injective for each coherent model M. Moreover, the reflection condition (15) for R imposes that $X_R \subseteq X_M\times X_N$ is the restriction of $Y_R \subseteq Y_M\times Y_N$ via $f_M\times f_N$ . We obtain that $ev_{({\mathbf {E}}, \mathcal {M})}(f)$ is in the right class of $\mathbf { Rel^{G({\mathbf {E}}, \mathcal {M})}}$ .

$\boldsymbol {ev}_{({{\mathbf{E}}}, \mathcal {M})}$ preserves the modality. One side of the preservation condition is (16), the other side is a converse of (16), namely

(17) $$ \begin{align} a\in \lbrack {\kern-1.3pt}\lbrack \Diamond S\rbrack {\kern-1.3pt}\rbrack_M ~~\Rightarrow~~{\text{there are}}~N, b, R~\mathrm{s.t.}~ aRb ~\mathrm{and}~b\in \lbrack {\kern-1.3pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack_N \end{align} $$

for every M, X, $S\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X)$ and $a \in X_M$ . To prove it, pick such $a\in \lbrack {\kern-1pt}\lbrack \Diamond S\rbrack {\kern-1.3pt}\rbrack _M$ and consider the following subsets of $\operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X)$ :

(18) $$ \begin{align} \Gamma~=~\{ S\}, \qquad \Delta~=~\{ T\mid a\not\in \lbrack {\kern-1.3pt}\lbrack \Diamond T\rbrack {\kern-1.3pt}\rbrack_M\}. \end{align} $$

Clearly $\Gamma $ is closed under finite meets and $\Delta $ is closed under finite joins (because $\Diamond $ commutes with joins) and downward-closed (because $\Diamond $ preserves the lattice ordering); in addition $\Gamma \cap \Delta =\emptyset $ since $a\in \lbrack {\kern-1pt}\lbrack \Diamond S\rbrack {\kern-1.3pt}\rbrack _M$ . By Proposition (4.2), there are a model N and $b\in X_N$ realizing a prime filter extending $\Gamma $ and disjoint from $\Delta $ . This means in particular that $b\in \lbrack {\kern-1pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack _N$ and that the family of relations ${R=\{ R(Y)\subseteq Y_M\times Y_N\}_Y}$ containing just the singleton pair $(a, b) \in R(X)$ satisfy condition (16). Now it is sufficient to apply Lemmas 4.8 and 4.9.

5 Saturation

Saturation aspects must be carefully addressed in a modal category $({\mathbf {E}}, \mathcal {M})$ . Here is a typical saturation problem:

Suppose that $R\hookrightarrow X\times Y$ in ${\mathcal {M}}$ is functional (i.e., that it satisfies the axioms of a functional relation in the Lawvere doctrine $\operatorname {\mathrm {Sub}}_{\mathcal {M}}$ of Section 2.2) and that it is ‘continuous’ in the obvious sense that ‘inverse image along R’ satisfies condition (5). Is R necessarily the graph of an arrow $X\longrightarrow Y$ ?

In general, there is no reason for the answer to be positive. Analogous problems arise concerning the notion of a subspace and of an isomorphism. We shall formulate below all the related saturation principles and then we shall prove that they are all equivalent, giving rise to a stronger notion of a ‘saturated’ modal category.

Let us start by analyzing the notion of a subspace, i.e., of an arrow $m:A \hookrightarrow X$ belonging to $ \mathcal {M}$ . We know that m must satisfy condition (6). We wonder what happens if a converse is satisfied: namely, is it sufficient that (6) holds for all ${S\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(A)}$ for a monomorphism m to be in $ \mathcal {M}$ ? If we look at our reference semantic category ${\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}}$ , we see that this is not the case, in the sense that we must make the condition ‘stable’ by adding a parameter Z for this to be true. This is similar to what happens for instance in Lawvere’s doctrines [30, 31] for the substitutivity axiom for equality: a parameter must be added to formulate it correctly. We say that a mono $m:A \hookrightarrow X$ is a brittle subspace iff for all Z, for all $S\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(A\times Z)$ we have that

(19) $$ \begin{align} \Diamond S=(m\times 1_Z)^* \Diamond\, \exists_{m\times 1_Z} \,S. \end{align} $$

In $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ , the ${\mathcal {M}}$ -subobjects can then be characterized as the brittle subspaces. ³ We can now formulate the following.

Subspace saturation principle (SS). A modal category satisfies the subspace saturation principle iff every brittle subspace is a subspace (i.e., it belongs to $ \mathcal {M}$ ).

Now, let us analyze functional relations: we define them to be the $R\buildrel {r}\over \hookrightarrow X\times Y$ in $ \mathcal {M}$ such that the composite map

$$ \begin{align*}R \buildrel{r}\over{\hookrightarrow} X\times Y \buildrel{\pi_X} \over \longrightarrow X \end{align*} $$

is a monomorphism and is also in ${\mathcal {M}}^\bot $ (we shall see below how to reformulate this and other similar conditions at the doctrinal level, so as to be able to use the internal language of doctrines in computations).

A continuous relation on the other hand is some $R\buildrel {r}\over \hookrightarrow X\times Y$ in $ \mathcal {M}$ such that for all Z, for all $S\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(Y\times Z)$ we have that the following inequality holds in $Sub_{ \mathcal {M}}(X\times Y\times Z)$ :

(20) $$ \begin{align} \Diamond \exists_{\pi_{X\times Z}}( \pi_{X\times Y}^*(R)\wedge \pi_{Y\times Z}^*(S)) \leq \exists_{\pi_{X\times Z}}( \pi_{X\times Y}^*(R)\wedge \pi_{Y\times Z}^*(\Diamond S)). \end{align} $$

If, in general, we abbreviate $\exists _{\pi _{X\times Z}}( \pi _{X\times Y}^*(R)\wedge \pi _{Y\times Z}^*(T))$ (for $T\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(Y\times Z)$ ) as $R\circ T$ , the above continuity condition can be written simply as

(21) $$ \begin{align} \Diamond (R\circ S) \leq R\circ (\Diamond S). \end{align} $$

We say that $R\in Sub_ {\mathcal {M}}(X\times Y)$ is a brittle morphism iff it is functional and continuous. We can now formulate the following.

Functional saturation principle (FS). A modal category satisfies the functional saturation principle iff every brittle morphism $R\hookrightarrow X\times Y$ is the graph of an arrow, meaning that there is $h: X\longrightarrow Y$ such that we have $(1_X, h)\simeq R$ in $\operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X\times Y)$ .

Finally, let us analyze the case of isomorphisms. We say that $h: X\longrightarrow Y$ is open iff it satisfies the condition

$$ \begin{align*}h^*(\Diamond S) \leq \Diamond (h^* S) \end{align*} $$

for every $S\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(Y)$ ; it is stably open iff $h\times 1_Z$ is open for every Z. We say that h is a brittle isomorphism iff it is mono, it is in $ \mathcal {M}^\bot $ and it is also stably open.

We can now formulate the following.

Isomorphism saturation principle (IS). A modal category satisfies the isomorphism saturation principle iff every brittle isomorphism is an isomorphism.

Before attacking the proof that the above saturation principles are all equivalent, let us make a practical remark. In concrete computations, it is convenient to rely on the doctrinal formalism and work in the underlying Lawvere doctrine of a modal category (see Proposition 2.7). To this aim it is important to know that most notions concerning an f-coherent category $({\mathbf {E}}, \mathcal {M})$ have equivalent reformulations at the doctrinal level in terms of subobjects inclusions. In particular, the fact that an arrow $f:X\longrightarrow Y$ belongs to ${\mathcal {M}}^\bot $ can be rewritten as $\top \leq \exists _f(\top )$ and the fact that f is mono can be rewritten as $(f\times f)^*(=_Y) \leq (=_X)$ , where $=_X, =_Y$ are the equalities over X and Y—recall that the diagonals are in $ \mathcal {M}$ , being the equalizers of the two projections. Thus for instance, we can establish that $f^*$ is injective (with $\exists _f$ as left inverse) when $f\in {\mathcal {M}}^\bot $ , that $f^*$ is surjective (with $\exists _f$ as right inverse) when f is mono, that $\exists _f$ and $f^*$ are inverse to each other when $f\in {\mathcal {M}}^\bot $ is also mono, etc. We shall refer to these kinds of arguments as ‘first-order reasoning’ below (such reasoning can be profitably worked off using the internal language of Lawvere doctrines).

Theorem 5.1. The above saturation principles are all equivalent in a modal category $({\mathbf {E}}, {\mathcal {M}})$ .

Proof. (SS $\Rightarrow $ IS): Suppose that $h:X\longrightarrow Y$ is in ${\mathcal {M}}^\bot $ , is mono and stably open. Take Z and $S\in \operatorname {\mathrm {Sub}}_ {\mathcal {M}} (X\times Z)$ ; we have

$$ \begin{align*}\Diamond S ~=~\Diamond (h\times 1_Z)^* \exists_{h\times 1_Z} S~=~(h\times 1_Z)^*\Diamond \exists_{h\times 1_Z} S \end{align*} $$

by first-order reasoning and stable openness. Thus $h\in {\mathcal {M}}$ by (SS) and consequently h is iso.

(IS $\Rightarrow $ SS): Let $m:A\longrightarrow X$ be a brittle subspace; let us factor it as $A \buildrel q\over \longrightarrow Q \buildrel i \over \longrightarrow X$ with $i\in \mathcal {M}$ and $q\in {\mathcal {M}}^\bot $ . We show that q is stably open (so that (IS) applies, q is iso and then $m\in \mathcal {M}$ ). Take Z and $S\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Q\times Z)$ ; we have by (6)

(first-order reasoning) $$ \begin{align} (q\times 1_Z)^*\Diamond S &= (q\times 1_Z)^*(i\times 1_Z)^*\Diamond \exists_{i\times 1_Z} S \nonumber\\&= (m\times 1_Z)^*\Diamond \exists_{i\times 1_Z} S \nonumber\\&= (m\times 1_Z)^*\Diamond \exists_{i\times 1_Z} \exists_{q\times 1_Z} (q\times 1_Z)^* S \end{align} $$

(by the hypothesis on m) $$\begin{align} &= (m\times 1_Z)^*\Diamond \exists_{m\times 1_Z} (q\times 1_Z)^* S \ \ \ \qquad\qquad\quad \nonumber\\&= \Diamond (q\times 1_Z)^* S. \end{align} $$

This shows that $q\times 1_Z$ is open.

(FS $\Rightarrow $ IS): Let $h:X\longrightarrow Y$ be a brittle isomorphism; consider R defined as

$$ \begin{align*}R:= X\buildrel{(h, 1_X)}\over\longrightarrow Y\times X. \end{align*} $$

Notice that $\pi _Y\circ R=h$ is mono and belongs to ${\mathcal {M}}^\bot $ by our hypotheses on h, so R is a functional relation; if we manage to prove that R is also continuous, then by functional saturation there is $f:Y\longrightarrow X$ such that $R=(h,1_X) \simeq (1_Y,f)$ , which means that h is iso with inverse f.

Now notice that, by first-order reasoning, for every Z and $S\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X\times Z)$ we have that $R\circ S$ is equal to $\exists _{h\times 1_Z}(S)$ . As a consequence, the continuity of our R can be expressed as the “Barcan formula” for $h\times 1_Z$

$$ \begin{align*}\Diamond \exists_{h\times 1_Z}S \leq \exists_{h\times 1_Z} \Diamond S, \end{align*} $$

which is an immediate consequence of the stable openness of h and the fact that h (and so also $h\times 1_Z$ ) satisfies the hypotheses of Lemma 5.2 below.

(IS $\Rightarrow $ FS): Let $R\buildrel {r}\over \hookrightarrow X\times Y$ be functional and continuous. We show that $R\buildrel {r}\over \hookrightarrow X\times Y \buildrel {\pi _X}\over \longrightarrow X$ is stably open: then (IS) applies and $\pi _X\circ r$ has an inverse $h:X\longrightarrow R$ witnessing the isomorphism of R with the graph of $\pi _Y\circ r\circ h$ . In order to show that $\pi _X\circ r$ is stably open, we use Lemma 5.2 and prove the “Barcan formula” for $(\pi _Y\circ r) \times 1_Z$ for every Z. To this aim, let $S\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(R\times Z)$ ; we have

$$ \begin{align*}\Diamond \exists_{(\pi_X\circ r)\times 1_Z} S~=~ \Diamond \exists_{\pi_{X\times Z}}\exists_{r\times 1_Z} S~ =~ \Diamond \exists_{\pi_{X\times Z}} (\pi_{X\times Y}^*R\wedge \exists_{r\times 1_Z} S) \end{align*} $$

by first-order reasoning and

$$ \begin{align*}\Diamond \exists_{\pi_{X\times Z}} (\pi_{X\times Y}^*R\wedge \exists_{r\times 1_Z} S) ~\leq~ \exists_{\pi_{X\times Z}} (\pi_{X\times Y}^*R\wedge \Diamond \exists_{r\times 1_Z} S) \end{align*} $$

by Lemma 5.3 below. We now continue by first-order reasoning ⁴ and obtain

$$ \begin{align*}\exists_{\pi_{X\times Z}} (\pi_{X\times Y}^*R\wedge \Diamond \exists_{r\times 1_Z} S) = \exists_{\pi_{X\times Z}} \exists_{r\times 1_Z}({r\times 1_Z})^*\Diamond \exists_{r\times 1_Z} S, \end{align*} $$

which is equal to $\exists _{(\pi _X\circ r)\times 1_Z}\Diamond S$ , as required.

Lemma 5.2. Suppose that $h\in {\mathcal {M}}^\bot $ is also mono. Then h is open iff it satisfies the “Barcan formula”, namely

(22) $$ \begin{align} \Diamond \exists_{h} S \leq \exists_{h} \Diamond S. \end{align} $$

for every $ \mathcal {M}$ -subobject S of the domain of h.

Proof. Trivial, by first-order reasoning, taking in mind that $\exists _h$ is the inverse correspondence of $h^*$ .

Lemma 5.3. Suppose that $R\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}}(X\times Y)$ is continuous; take $\tilde Z$ and ${U\in \operatorname {\mathrm {Sub}}_{{\mathcal {M}}}(Y\times X\times {\tilde Z})}$ . We have

(23) $$ \begin{align} \Diamond \exists_{\pi_{X\times \tilde Z}} (\pi_{X\times Y}^*R\wedge U) ~\leq~ \exists_{\pi_{X\times \tilde Z}} (\pi_{X\times Y}^*R\wedge \Diamond U). \end{align} $$

Proof. Observe that, by first-order reasoning or directly by applying the Beck–Chevalley condition to a suitable pullback square, we have

(24) $$ \begin{align} \exists_{\pi_{X\times \tilde Z}}((\pi_{X\times Y}^*R)\wedge U) = (\Delta_X \times 1_{\tilde Z})^*(R\circ U) \text{.} \end{align} $$

Thus we get

(by continuity of R) $$\begin{align} \Diamond \exists_{\pi_{X\times \tilde Z}} (\pi_{X\times Y}^*R\wedge U) &= \Diamond (\Delta_X \times 1_{\tilde Z})^*(R\circ U) \nonumber\\ &\leq (\Delta_X \times 1_{\tilde Z})^* \Diamond (R\circ U) \nonumber\\ &\leq (\Delta_X \times 1_{\tilde Z})^* (R \circ \Diamond U) \\ &= \exists_{\pi_{X\times \tilde Z}} (\pi_{X\times Y}^*R\wedge \Diamond U).\nonumber\end{align} $$

Definition 5.4. We say that a modal category is saturated iff it satisfies one of (hence all) the saturation principles (SS), (FS), and (IS).

An effect of the saturation condition is that $ \mathcal {M}$ -conservative embeddings coincide with conservative embeddings.

Proposition 5.5. Let $\Phi :({\mathbf {E}}_1, \mathcal {M}_1)\longrightarrow ({\mathbf {E}}_2, \mathcal {M}_2)$ be a modal morphism and let $({\mathbf {E}}_1, \mathcal {M}_1)$ be a saturated modal category; then $\Phi $ is $ \mathcal {M}$ -conservative iff it is conservative.

Proof. The right to left direction is trivial. For the converse, let $\Phi $ be $ \mathcal {M}$ -conservative. Notice that, for $S,T\in \operatorname {\mathrm {Sub}}_{ \mathcal {M}_1}(X)$ , we have that “ $S\leq T$ and $\Phi (S)\simeq \Phi (T)$ imply $S\simeq T$ ”: this is because of Lemma 2.4 and $ \mathcal {M}$ -conservativity. Suppose now that m is mono and that $\Phi (m)$ is iso. Then $m\in {\mathcal {M}}^\bot $ because the inequality $\exists _m (\top ) \leq \top $ becomes an equality in ${\mathbf {E}}_2$ , so that $\exists _m (\top ) \simeq \top $ . For the same reason, $m\times 1_Z$ is open for all Z because $\Phi (m)\times 1_{\Phi (Z)}$ is open being an iso. Thus m is a brittle isomorphism and so it is iso since $({\mathbf {E}}_1, \mathcal {M}_1)$ is saturated.

As an immediate corollary, we can strengthen our representation theorem.

Theorem 5.6. For every saturated small modal category $({\mathbf {E}}, \mathcal {M})$ , the evaluation functor

$$ \begin{align*}ev_{({\mathbf{E}}, \mathcal{M})}: ({\mathbf{E}}, \mathcal{M}) \longrightarrow \mathbf{Rel^{G({\mathbf{E}}, \mathcal{M})}} \end{align*} $$

is a conservative embedding of modal categories.

6 Quotients and disjoint unions

Being a quasi-topos, $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ is a coherent category [27]. This fact distinguishes it from other semantic modal categories (e.g., topological spaces and preordered sets are not regular categories). In addition, in $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ sums exist, are disjoint and pullback stable. However $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ fails to be a pretopos because only equivalence relations which are regular monos are effective. In this section we show how to enrich modal categories with quotients and disjoint unions, in order to eventually define the notion of a modal quasi-pretopos. We limit ourselves to Boolean modal categories in order to avoid technical complications, because the correct condition for the “quotients” in general (replacing (25) below) is more involved and not even algebraic.

Definition 6.1. A Boolean modal category with quotients is a Boolean modal category $({\mathbf {E}}, {\mathcal {M}})$ equipped with a further distinguished class $\mathcal {Q}$ of maps called quotients and which satisfy the following conditions:

1. 1. Each arrow $q \in \mathcal {Q}$ is surjective in the internal logic, meaning that $\mathcal {Q} \subseteq {\mathcal {M}}^\bot $ .

2. 2. If $q : X\longrightarrow Y$ is a quotient, then for every Z and for every $S\in \operatorname {\mathrm {Sub}}_{{\mathcal {M}}}(Y\times Z)$ we have

   (25) $$ \begin{align} \Diamond S \leq \exists_{q\times 1_Z} \Diamond (q\times 1_Z)^*S. \end{align} $$

A morphism of Boolean modal categories with quotients is a modal functor which sends quotients to quotients.

For instance, if $\mathbf {G}$ is a graph, then $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ is equipped with a canonical quotient structure by declaring that $f : X\longrightarrow Y$ is a quotient when

• • $f_\alpha : X_\alpha \longrightarrow Y_\alpha $ is surjective for every $\alpha \in \mathbf {G}$ and

• • if k is an arrow of $\mathbf {G}$ and $s~k~t$ in Y, then there are $s', t' \in X$ with $s'~k~t'$ and $f(s') = s$ and $f(t') = t$ .

Thus in $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ (but not in $\mathbf {Top}$ ) quotients turn out to be just regular epi.

Let us fix a small Boolean modal category with quotients $({\mathbf {E}}, {\mathcal {M}},\mathcal {Q})$ . Recall from Section 4 the definition of a modal transformation between two coherent models M, N of $({\mathbf {E}}, {\mathcal {M}},\mathcal {Q})$ : it is a family of relations $R=\{R(X) \subseteq M(X) \times N(X)\}_{X\in {\mathbf {E}}}$ satisfying conditions (13)–(16). We say that R respects quotients whenever

(26) $$ \begin{align} a~R~b \implies \exists a',b' : q(a')=a ~\&~ q(b')=b ~\&~ a'~R~b' \end{align} $$

for any quotient q.

To continue, we need to introduce $\lambda $ -saturated models (this is a standard notion in model theory [10] that we are going to reformulate in our context). Given ${X,Y \in {\mathbf {E}}}$ , we will denote by $\pi _Y : X\times Y \longrightarrow Y$ the canonical projection. Let $M : {\mathbf {E}} \longrightarrow \mathbf {Set}$ be a coherent model; take a lex subfunctor $\mathcal {S}\subseteq M$ and some object X in ${\mathbf {E}}$ . An X-pretype over ${\mathcal S}$ is a collection $\Theta $ of subsets $\Theta _a \subseteq \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times Z)$ indexed by $a\in {\mathcal S}(Z)$ such that:

1. (PT0) $\Theta _a$ is a prefilter, i.e., a non empty subset of $\operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times Z)$ such that for each $A, B \in \Theta _a$ , there is $C\in \Theta _a$ such that $C \leq A \land B$ . Thus, the filter generated by $\Theta _a$ coincides with the up-set generated by $\Theta _a$ , denoted ${\uparrow }\Theta _a$ .

2. (PT1) If $(a',a)\in {\mathcal S}(Z'\times Z)$ and $A\in \Theta _a$ then $(1_X\times \pi _Z)^* A\in {\uparrow }\Theta _{(a',a)}$ .

3. (PT2) If $A\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Z)$ , $\pi _Z:X\times Z \longrightarrow Z$ is the projection, and $a\in \lbrack {\kern-1pt}\lbrack A\rbrack {\kern-1.3pt}\rbrack _M$ , then $\pi _Z^*A \in {\uparrow } \Theta _a$ .

We say that $c \in M(X)$ realizes $\Theta $ if for all Z, $a\in \mathcal {S}(Z)$ and $A\in \Theta _a$ , we have that $(c,a)\in \lbrack {\kern-1pt}\lbrack A\rbrack {\kern-1.3pt}\rbrack _M $ . We say that $\Theta $ is consistent when ${\uparrow }\Theta _a$ never contains $\bot $ . Given a regular cardinal $\lambda $ , we say that M is $\lambda $ -saturated iff for every lex subfunctor $\mathcal {S}$ such that the cardinality of $\sum _{Z\in {\mathbf {E}}} \mathcal {S}(Z)$ is less than $\lambda $ , every M-consistent X-pretype over $\mathcal S$ is realized in M.

Remark 6.2. Given a pretype $\Theta $ , to facilitate the comparison with the definition from model theory textbooks, it might be useful to write $A\in \Theta _a$ as $A(x^X,a)\ \in \Theta $ . Thus the elements of $\Theta $ can be viewed as formulae with a free variable $x^X$ of type X in the expanded language with constants from the support of $\mathcal S$ . Seen in this way, condition (PT1) becomes tautological and condition (PT2) just says that if $A(a)$ is true in M then it belongs to $\Theta $ . Similarly, c realizes $\Theta $ iff for all $A(x^X,a)\ \in \Theta $ we have that $A(c,a)$ is true in M. The notation $A(x^X,a)\ \in \Theta $ is more transparent, notice however that it should be clear that this notation does not refer to an expansion of the modal language with constants. Rather we have pairs given by formulae in the original modal language and constants naming elements from the model. As a consequence, if A is $\Diamond A'$ , then in $\Diamond A'(x^X,a)$ the ‘constant’ a stays outside the modal operator. In other terms, we expanded not the modal language but the classical first-order language used to encode the modal language (see Section 7 below for details).

Remark 6.3. If (PT2) is satisfied, in (PT0) there is no need to ask that $\Theta _a$ is not empty ( $\pi _Z^*\top $ will be in $\Theta _a$ anyway). In a positive context, the axiom (PT2) has to be transformed into an equivalence: $a\in \lbrack {\kern-1pt}\lbrack A\rbrack {\kern-1.3pt}\rbrack _M$ iff $\pi _Z^*A \in {\uparrow } \Theta _a$ . In a Boolean context, the left-to-right direction is sufficient because then $a\not \in \lbrack {\kern-1pt}\lbrack A\rbrack {\kern-1.3pt}\rbrack _M$ implies $\lnot \pi _Z^*A \in {\uparrow } \Theta _a$ and if also $\pi _Z^*A \in {\uparrow } \Theta _a$ , then $\Theta $ is in fact inconsistent.

Remark 6.4. It can be shown that (PT1) can be equivalently stated using arrows which are not necessarily projections as follows: if $f:Z'\longrightarrow Z$ is such that $f_M(a')=a$ and $A\in \Theta _a$ then $(1_X\times f)^* A\in {\uparrow }\Theta _{a'}$ . This suggests a slightly different (but substantially equivalent) definition of a pretype, where one takes as $\Theta $ the filtered colimit of the $\Theta _a$ (here the filtered colimit is taken over the opposite of the category of elements of $\mathcal {S}$ , which is co-filtered because $\mathcal S$ is lex).

In the following we will be interested in $\mathrm {card}{\,{\mathbf {E}}}^+$ -saturated models, where $\mathrm {card}{\,{\mathbf {E}}}^+$ is the successor cardinal of the cardinality $\mathrm {card}{\,{\mathbf {E}}}$ of the set of arrows of ${\mathbf {E}}$ . Since applying ultrapower modulo a suitable ultrafilter to a model M makes it $\mathrm {card}{\,{\mathbf {E}}}^+$ -saturated [10, Theorems 6.1.4 and 6.1.8], it is clear that there are enough $\mathrm {card}{\,{\mathbf {E}}}^+$ -saturated models, e.g., in the sense that Proposition 4.2 applies not only to plain coherent models but also to $\mathrm {card}{\,{\mathbf {E}}}^+$ -saturated ones.

Lemma 6.5. Let $M,N$ be two $\mathrm {card}{\,{\mathbf {E}}}^+$ -saturated coherent models of the Boolean modal category with quotients $({\mathbf {E}}, {\mathcal {M}},\mathcal {Q})$ and let $R\subseteq M\times N$ such that $\mathrm {card}{\,R} \leq \mathrm {card}{\,{\mathbf {E}}}$ . Suppose that R satisfies (13) and (16). Let $q : X\twoheadrightarrow Y$ be a quotient in $\mathcal {Q}$ . Let $(a,b) \in Y_M \times Y_N$ such that $a\,R\,b$ . Then R is contained in some $R' \subseteq M\times N$ satisfying (13) and (16) and containing some $(a',b') \in X_M\times X_N$ such that $q(a') = a$ and $q(b') = b$ . Moreover, $\mathrm {card}{\,R'} \leq \mathrm {card}{\,{\mathbf {E}}}$ .

Proof. We find first $a'$ , $b'$ and finally $R'$ .

Construction of $\mathbf {a'}$ . Let $\mathcal S$ be a lex subfunctor of M such that $\sum _{Z\in {\mathbf {E}}} \mathcal {S}(Z)$ has cardinality at most $\mathrm {card}{\,{\mathbf {E}}}$ and comprises all c such that there is $c'$ with $cR c'$ . Given $f:Z\longrightarrow V$ , $A\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times Z)$ and $B \in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times V)$ we write

$$\begin{align*}F(f,A,B) = (q \times 1_Z)^*(A) \land (1_X\times f)^*\Diamond(q\times 1_V)^*(B) \text{.} \end{align*}$$

For $c\in \mathcal {S}(Z)$ , let $\Theta _c$ be the set of the subobjects $S\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times Z)$ such that there exist $f:Z\longrightarrow V$ , $d\in V_N$ , $A\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times Z)$ , $B\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times V)$ such that $f(c)Rd$ , $S=F(f,A,B)$ , $(a,c)\in \lbrack {\kern-1pt}\lbrack A\rbrack {\kern-1.3pt}\rbrack _M$ and $(b,d)\in \lbrack {\kern-1pt}\lbrack B\rbrack {\kern-1.3pt}\rbrack _N$ . This defines an X-pretype $\Theta $ over $\mathcal {S}$ because:

1. 1. $\Theta _c$ is a prefilter on $\operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times Z)$ and (PT0) is satisfied: in fact, if we have $f_i:Z\longrightarrow V_i$ , $d_i\in N_{V_i}$ , $A_i\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times Z)$ , $B_i\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times V_i)$ such that $f_i(c)Rd_i$ , $S_i= F(f_i, A_i,B_i)$ , $(a,c)\in \lbrack {\kern-1pt}\lbrack A_i\rbrack {\kern-1.3pt}\rbrack _M$ and $(b,d_i)\in \lbrack {\kern-1pt}\lbrack B_i\rbrack {\kern-1.3pt}\rbrack _N$ (for $i=1,2)$ , then taking $V:=V_1\times V_2$ , $f:=(f_1,f_2)$ ,

   $$ \begin{align*}~A:=A_1\wedge A_2,~ B:=(\pi_{Y\times V_1})^*B_1\wedge (\pi_{Y\times V_2})^*B_2,\end{align*} $$
   we have by continuity of pullbacks along projections
   $$ \begin{align*}F(f,A,B) \leq F(f_1,A_1,B_1) \wedge F(f_2,A_2,B_2) \end{align*} $$
   and moreover $f(c)=(f_1(c),f_2(c))R(d_1,d_2)$ by (13), $(a,c)\in \lbrack {\kern-1pt}\lbrack A\rbrack {\kern-1.3pt}\rbrack _M$ and $(b,d_1,d_2)\in \lbrack {\kern-1pt}\lbrack B\rbrack {\kern-1.3pt}\rbrack _N$ .

2. 2. If $(c',c)\in {\mathcal S}(Z'\times Z)$ and $F(f,A,B)\in \Theta _c$ with $f:Z\longrightarrow V$ and the required data, we have that $(1_X \times \pi _Z)^* F(f,A,B) = F(\pi _Z f , \pi _{Y\times Z}^* A,B)\in \Theta _{c,c'}$ with the obvious data, so that (PT1) is satisfied.

3. 3. If $C \in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Z)$ and $c\in \lbrack {\kern-1pt}\lbrack C\rbrack {\kern-1.3pt}\rbrack _M$ , then $\pi _Z^*C \in {\uparrow } \Theta _c$ : to see this, take ${f: Z\longrightarrow \mathbf {1}}$ , $A:=\pi _Z^*C,$ and $B=\top $ . Thus (PT2) is satisfied.

The consistency of $\Theta $ is established as follows: if we have (under the above mentioned data)

$$ \begin{align*}F(f,A,B) = (q \times 1_Z)^*(A) \land (1_X\times f)^*\Diamond(q\times 1_V)^*(B) \leq \bot \end{align*} $$

then applying $\exists _{q\times 1}$ to this inequality, using the Frobenius condition, the Beck–Chevalley condition and finally Definition 6.1, we get

$$ \begin{align*}A \land (1_Y\times f)^*\Diamond B \leq \bot, \end{align*} $$

which is impossible since $(a,c)\in \lbrack {\kern-1pt}\lbrack A\rbrack {\kern-1.3pt}\rbrack _M$ , $(b,d)\in \lbrack {\kern-1pt}\lbrack B\rbrack {\kern-1.3pt}\rbrack _N$ , $aRb$ , $f(c) R d,$ and so $(a,f(c))\in \lbrack {\kern-1pt}\lbrack \Diamond B\rbrack {\kern-1.3pt}\rbrack _M$ , because R satisfies (13) and (16).

Thus $\Theta $ is realized in M by some $a'\in M(X)$ . Taking in particular f to be the identity arrow and A to be the equality over Y in the above data, we see that $ q(a')=a$ and that whenever we have $c R d$ and $(b,d)\in \lbrack {\kern-1pt}\lbrack B\rbrack {\kern-1.3pt}\rbrack _N$ we have also $(a',c)\in \lbrack {\kern-1pt}\lbrack \Diamond (q\times 1)^* B\rbrack {\kern-1.3pt}\rbrack _M$ .

Construction of $\mathbf {b'}$ . Let now $\mathcal S'$ be a lex subfunctor of N such that $\sum _{Z\in {\mathbf {E}}} \mathcal {S}'(Z)$ has cardinality at most $\mathrm {card}{\,{\mathbf {E}}}$ and comprises all d such that there is c with $cR d$ . Given $f:Z\longrightarrow V$ and $A\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times V)$ , $B\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times Z)$ we write

$$\begin{align*}G(f,A,B) = (1_X \times f)^*(A) \land (q\times 1_Z)^*(B) \text{.} \end{align*}$$

For $d\in \mathcal {S}'(Z)$ , let $\Psi _d$ be the set of subobjects $S\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times Z)$ such that there exist $f:Z\longrightarrow V$ , $c\in V_M$ , $A\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times V)$ , $B\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times Z)$ such that $cRf(d)$ , $S=G(f,A,B)$ , $(a',c)\in \lbrack {\kern-1pt}\lbrack \Box A\rbrack {\kern-1.3pt}\rbrack _M$ and $(b,d)\in \lbrack {\kern-1pt}\lbrack B\rbrack {\kern-1.3pt}\rbrack _N$ . This defines an X-pretype $\Psi $ over $\mathcal {S}'$ because:

1. 1. $\Psi _d$ is a prefilter on $\operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times Z)$ and (PT0) is satisfied: in fact, if we have $f_i:Z\longrightarrow V_i$ , $c_i\in M_{V_i}$ , $A_i\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(X\times V_i)$ , $B_i\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Y\times Z)$ such that $c_iRf_i(d)$ , $S_i=G(f_i,A_i,B_i)$ , $(a',c_i)\in \lbrack {\kern-1pt}\lbrack \Box A_i\rbrack {\kern-1.3pt}\rbrack _M$ and $(b,d)\in \lbrack {\kern-1pt}\lbrack B_i\rbrack {\kern-1.3pt}\rbrack _N$ (for $i=1,2$ ), then taking $f:=(f_1,f_2)$ ,

   $$ \begin{align*}~A:=(\pi_{X\times V_1})^*A_1\wedge (\pi_{X\times V_2})^*A_2,~ B:=B_1\wedge B_2,\end{align*} $$
   we have $G(f,A,B)\leq G(f_1,A_1,B_1) \wedge G(f_2,A_2,B_2)$ and moreover $(c_1,c_2)Rf(d)$ , $(b,d)\in \lbrack {\kern-1pt}\lbrack B\rbrack {\kern-1.3pt}\rbrack _N$ and finally $(a',c_1,c_2)\in \lbrack {\kern-1pt}\lbrack \Box A\rbrack {\kern-1.3pt}\rbrack _M$ because
   $$ \begin{align*}(\pi_{X\times V_1})^* \Box A_1\wedge (\pi_{X\times V_2})^*\Box A_2 \leq \Box ((\pi_{X\times V_1})^*A_1\wedge (\pi_{X\times V_2})^*A_2) \end{align*} $$
   by continuity.

2. 2. If $(d',d)\in {\mathcal S}(Z'\times Z)$ and $G(f,A,B)\in \Psi _d$ with $f:Z\longrightarrow V$ and the required data, then $(1_X\times \pi _Z)^*G(f,A,B) = G(\pi _Z f,A,\pi _{Y\times Z}^* B)\in \Psi _{d,d'}$ with the obvious data, so that (PT1) is satisfied.

3. 3. If $D {\kern-1pt}\in{\kern-1pt} \operatorname {\mathrm {Sub}}_{\mathcal {M}}(Z)$ and $d{\kern-1pt}\in{\kern-1pt} \lbrack {\kern-1pt}\lbrack D\rbrack {\kern-1.3pt}\rbrack _M$ , then $\pi _Z^*D \in {\uparrow } \Psi _d$ : to see this, take ${f {\kern-1pt}:{\kern-1pt} Z \longrightarrow \mathbf {1}}$ , c the unique element of $M_{\mathbf {1}}$ , $A=\top $ and $B=\pi _Z^*D$ . Thus (PT2) is satisfied.

The consistency of $\Psi $ is established as follows: if we have (under the above mentioned data)

$$\begin{align*}(1_X \times f)^*(A) \land (q\times 1_Z)^*(B) \leq \bot \end{align*}$$

then applying $\exists _{1_X\times f}$ and using the Frobenius and the Beck–Chevalley conditions, we get

$$ \begin{align*}A\wedge (q\times 1_V)^*\exists_{1_Y\times f} B\leq \bot \end{align*} $$

and hence

$$ \begin{align*}\Box A\wedge \Diamond (q\times 1_V)^*\exists_{1_Y\times f} B\leq \bot.\end{align*} $$

From $(b,d)\in \lbrack {\kern-1pt}\lbrack B\rbrack {\kern-1.3pt}\rbrack _N$ we get $(b,f(d))\in \lbrack {\kern-1pt}\lbrack \exists _{1_Y\times f} B\rbrack {\kern-1.3pt}\rbrack _N$ and from $cRf(d)$ and the construction of $a'$ , we obtain $(a',c)\in \lbrack {\kern-1pt}\lbrack \Diamond (q\times 1_V)^*\exists _{1_Y\times f} B\rbrack {\kern-1.3pt}\rbrack _M$ . This, together with $(a',c)\in \lbrack {\kern-1pt}\lbrack \Box A\rbrack {\kern-1.3pt}\rbrack _M$ yields a contradiction.

Thus $\Psi $ is consistent and realized in N by some $b'$ . Taking in particular B to be the equality over Y and f to be the identity in the above data, we see that $ q(b')=b$ and $(b',d) \in \lbrack {\kern-1pt}\lbrack A\rbrack {\kern-1.3pt}\rbrack _N$ implies $(a',c)\in \lbrack {\kern-1pt}\lbrack \Diamond A\rbrack {\kern-1.3pt}\rbrack _M$ whenever $cRd$ (remember that $\Diamond =\neg \Box \neg $ ).

Construction of $\mathbf {R'}$ . Let $R'$ be the family of relations defined as follows, for every Z and $c\in Z_M$ , $d\in Z_N$ :

$c R' d$ iff there exist $g:X\times Z'\longrightarrow Z, c'\in Z^{\prime }_M, d'\in Z^{\prime }_N$ such that $c' R d'$ and $c=g(a',c'), d=g(b',d')$ .

It is immediate to see that $R'$ satisfies conditions (13) and (16), so it is the desired $R'$ .

Lemma 6.6. Let $M,N$ be two $\mathrm {card}{\,{\mathbf {E}}}^+$ -saturated coherent models of a Boolean modal category with quotients $({\mathbf {E}}, {\mathcal {M}},\mathcal {Q})$ and let $R_0\subseteq M\times N$ such that $\mathrm {card}{\,R_0} \leq \mathrm {card}{\,{\mathbf {E}}}$ . Suppose that $R_0$ satisfies (13) and (16). Then $R_0$ is contained in some modal transformation $R' \subseteq M\times N$ satisfying also (26).

Proof. Pick a well ordering of the triples $(a,b,q)$ (where $q:X\longrightarrow Y$ is in $\mathcal {Q}$ , $a\in Y_M$ , $b\in Y_N$ and $aRb$ ) by some ordinal $\kappa $ (notice that the cardinality of $\kappa $ does not exceed $\mathrm {card}{\,{\mathbf {E}}}$ ). We define a sequence $(R_\xi \subseteq M\times N)_{\xi \leq \kappa }$ by:

• • $R_0$ is given;

• • $R_{\xi +1}$ is obtained by applying Lemma 6.5 to $R_\xi $ and the $\xi $ -th element of our well ordering;

• • $R_\lambda = \bigcup _{\xi <\lambda } R_\xi $ when $\lambda $ is a limit ordinal.

By induction, we have $\mathrm {card}{\,R_\xi } \leq \mathrm {card}{\,{\mathbf {E}}}$ for all $\xi $ . We then define $R^{\prime }_1$ as $\tilde {R_\kappa }$ , i.e., we apply the construction of Lemma 4.7 to ${R_\kappa }$ . By referring to the construction presented in Lemma 4.7, we see that $\mathrm {card}{\,R^{\prime }_1} \leq \mathrm {card}{\,{\mathbf {E}}}$ . Since $R_\kappa $ satisfies (13) and (16), we obtain that $R^{\prime }_1$ is a modal transformation satisfying all conditions (13)–(16) and also (26), but the latter only when $(a,b)$ is taken in $R_0 \subseteq R^{\prime }_1$ . To get the desired $R'$ it is now sufficient to repeat $\omega $ -times the above construction leading from $R_0$ to $R^{\prime }_1$ , and take the union.

Theorem 6.7. If $({\mathbf {E}}, {\mathcal {M}},\mathcal {Q})$ is a small Boolean modal category with quotients, then there is a graph $\mathbf {G}$ and a morphism ${\mathbf {E}} \longrightarrow \operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ of Boolean modal categories with quotients which is ${\mathcal {M}}$ -conservative.

Proof. Let $\lambda =\mathrm {card}{\,{\mathbf {E}}}^+$ be the successor of the cardinal $\mathrm {card}{\,{\mathbf {E}}}$ . Let $\mathbf {G}_\lambda ({\mathbf {E}})$ be the graph of coherent $\lambda $ -saturated models of the f-coherent category $({\mathbf {E}}, {\mathcal {M}})$ and modal transformations which respect quotients. We will show that the canonical evaluation functor

$$ \begin{align*}ev_{({\mathbf{E}}, {\mathcal{M}})} : {\mathbf{E}} \longrightarrow \operatorname{\mathrm{\mathbf{Rel}}}^{\mathbf{G}_\lambda({\mathbf{E}})}\end{align*} $$

is a morphism of Boolean modal categories. We know that it is ${\mathcal {M}}$ -conservative, since there are enough $\lambda $ -saturated models (see the extension of Proposition 4.2 to $\lambda $ -saturated models).

The only non-immediate thing that we must check is that whenever $M \in \mathbf {G}_\lambda ({\mathbf {E}})$ , for each $S \subseteq X$ in ${\mathbf {E}}$ and each $a \in \lbrack {\kern-1pt}\lbrack \Diamond S\rbrack {\kern-1.3pt}\rbrack _M$ there is some $N \in \mathbf {G}_\lambda ({\mathbf {E}})$ , a modal transformation $R \subseteq M\times N$ which respects quotients and some $b \in \lbrack {\kern-1pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack _N$ such that $(a,b) \in R$ . We start by building $N \in \mathbf {G}_\lambda ({\mathbf {E}})$ and $b \in X_N$ such that $b \in \lbrack {\kern-1pt}\lbrack S\rbrack {\kern-1.3pt}\rbrack _N$ and for any $S' \subseteq X$ such that $b \in \lbrack {\kern-1pt}\lbrack S'\rbrack {\kern-1.3pt}\rbrack _N$ we have $a \in \lbrack {\kern-1pt}\lbrack \Diamond S'\rbrack {\kern-1.3pt}\rbrack _M$ . The reasoning is exactly the same as in the case without quotients. Lemma 6.6 now shows that the pair $(a,b) \in X_M \times X_N$ is contained in a modal transformation $R \subseteq M\times N$ respecting quotients.

Definition 6.8. In a modal category $({\mathbf {E}}, {\mathcal {M}})$ , a disjoint union of two objects $A,B$ is an object $A+B$ endowed with two ${\mathcal {M}}$ -subobjects

$$ \begin{align*}\iota_A: A\hookrightarrow A+B, \qquad \iota_B: B\hookrightarrow A+B, \end{align*} $$

which are the Boolean complement of each other and such that

(27) $$ \begin{align} \Diamond A\leq A \qquad \text{and} \qquad \Diamond B\leq B \text{,} \end{align} $$

where the $\Diamond $ operator is taken inside of $A+B$ .

In $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ and in $\mathbf {Top}$ disjoint unions exist for every $A, B$ and are computed as in $\mathbf {Set}$ .

Proposition 6.9. The disjoint union injections $\iota _A: A\hookrightarrow A+B, \iota _B: B\hookrightarrow A+B$ are open, meaning that we have $\iota _A^* \Diamond S = \Diamond \iota _A^* S$ for all $S\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(A+B)$ and similarly for B.

Proof. For $S\in \operatorname {\mathrm {Sub}}_{\mathcal {M}}(A+B)$ , we have $S=S_1\vee S_2$ , with $S_1\leq A$ and ${S_2\leq B}$ . Then $\iota _A^* \Diamond S=\iota _A^* \Diamond S_1 \vee \iota _A^* \Diamond S_2=\iota _A^* \Diamond S_1$ by (27). On the other hand $\Diamond \iota _A^* S= \iota _A^*\Diamond \exists _{\iota _A}\iota _A^* S=\iota _A^* \Diamond S_1$ , by (6). The argument showing openness of $\iota _B$ is the same.

We now introduce Boolean modal quasi-pretoposes, but we warn the reader that the axioms for a Boolean modal quasi-pretopos acquire full transparent meaning only in the presence of saturation conditions (as we show below).

Definition 6.10. A Boolean modal quasi-pretopos is a Boolean modal category with quotients $({\mathbf {E}}, \mathcal {M}, \mathcal {Q})$ satisfying the following additional properties:

1. (i) every pair of objects $A, B \in {\mathbf {E}}$ has a disjoint union $A+B$ ;

2. (ii) every $R\in \operatorname {\mathrm {Sub}}_{{\mathcal {M}}}(X\times X)$ which is an equivalence relation is the kernel of some quotient $q : X\longrightarrow Q$ from $\mathcal {Q}$ .

A morphism of Boolean modal quasi-pretoposes is simply a morphism of modal categories with quotients. Note that such a morphism automatically sends disjoint unions to disjoint unions, since the conditions they must satisfy involve only coherent and modal operators.

$\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ is a Boolean modal quasi-pretopos (but $\mathbf {Top}$ is not because the quotient topology may fail to satisfy (25)). In $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ , disjoint unions are even unique modulo isomorphisms (in fact, as we will see below in Theorem 6.13, this is a consequence of saturation). The representation theorem extends to Boolean modal quasi-pretoposes.

Theorem 6.11. If $({\mathbf {E}}, {\mathcal {M}},\mathcal {Q})$ is a small Boolean modal quasi-pretopos, then there is a graph $\mathbf {G}$ and a morphism ${\mathbf {E}} \longrightarrow \operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ of Boolean modal quasi-pretoposes which is ${\mathcal {M}}$ -conservative.

Proof. Immediate by Theorem 6.7.

We now analyze some remarkable features of Boolean modal quasi-pretoposes, that can all be deduced from the above representation theorem.

Proposition 6.12. In a Boolean modal quasi-pretopos $({\mathbf {E}}, {\mathcal {M}},\mathcal {Q})$ , the class ${\mathcal {M}}$ is the class of regular monos.

Proof. Let $S\buildrel {s}\over \hookrightarrow X$ be in ${\mathcal {M}}$ . If we let $\iota _1, \iota _2$ be the two injections $X\rightrightarrows X+X$ , then $(s\iota _1, s\iota _2):S \hookrightarrow (X+X)^2$ is in ${\mathcal {M}}$ by Lemma 2.2 and its reflexive closure, i.e., its join in $\operatorname {\mathrm {Sub}}_{\mathcal {M}}((X+X)^2)$ with the diagonal, is an equivalence relation. Take its quotient $q: X+X\twoheadrightarrow Q$ and then the equalizer $E \buildrel {e}\over \hookrightarrow X$ of $\iota _1q$ and $\iota _2q$ . By the universal property, there is k such that $ke=s$ . Now $k\in {\mathcal {M}}$ as the first component of an arrow in ${\mathcal {M}}$ (see Lemma 2.4). However, it is not difficult to see that in every ${\mathcal {M}}$ -conservative embedding $\eta $ of $({\mathbf {E}}, {\mathcal {M}})$ into a category of the kind $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ , we have that $\eta (k)$ is iso (this is because coproducts and quotients of equivalence relations are computed set-wise in $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf { G}}$ ). Thus k is iso by Theorem 6.11 and so s is an equalizer.

In the hypothesis of saturation, more can be proved, because Theorem 6.11 supplies a conservative embedding (see Proposition 5.5).

We first make an observation concerning zero objects. A zero object is an object whose $ \mathcal {M}$ -subobjects lattice is a singleton; zero objects arise as minimum $ \mathcal {M}$ -subobjects. Notice that if O is a zero object and there is $f:X\longrightarrow O$ , then X is also a zero object (because from $\top \leq \bot $ in $\operatorname {\mathrm {Sub}}_{\mathcal {M}}(O)$ we can derive the same in $\operatorname {\mathrm {Sub}}_{\mathcal {M}}(X)$ ). Moreover there cannot be two different arrows $X\rightrightarrows O$ (because their equalizer must be the identity) and any arrow $X\longrightarrow O$ must be mono.

Theorem 6.13. Let $({\mathbf {E}}, {\mathcal {M}},\mathcal {Q})$ be a saturated Boolean modal quasi-pretopos. Then:

1. (i) disjoint unions are coproducts;

2. (ii) a zero object is an initial object;

3. (iii) $\mathcal {Q}$ is the set of regular epis;

4. (iv) regular epis are pullback stable (so ${\mathbf {E}}$ is a regular category).

Proof. (Ad (i)). Let

$$ \begin{align*}A\buildrel{\iota_A}\over \longrightarrow A+B \buildrel{\iota_B}\over\longleftarrow B \end{align*} $$

be a disjoint union and let

$$ \begin{align*}A\buildrel{f}\over \longrightarrow X \buildrel{g}\over\longleftarrow B \end{align*} $$

be further arrows as displayed. We must find a unique $h:A+B\longrightarrow X$ such that $h\circ \iota _A=f$ and $g\circ \iota _B=g$ . Uniqueness comes from the fact that the equalizer ${E\hookrightarrow A+B}$ of two such h’s must be equal to $\top $ , because it is easy to see, by first-order reasoning, that $E\wedge A=A$ and $E\wedge B=B$ . For existence, consider the relation $R \buildrel {r}\over {\hookrightarrow } (A+B)\times X$ given by

$$ \begin{align*}(\exists_{\iota_A\times 1_X}(f\times 1_X)^* =_X) \vee (\exists_{\iota_B\times 1_X}(g\times 1_X)^* =_X)~. \end{align*} $$

In the internal language of doctrines, this can be expressed by the formula

$$ \begin{align*}\exists a \,(\iota_A(a)=z \wedge f(a)=x) \vee \exists b \,(\iota_B(b)=z \wedge g(b)=x) \end{align*} $$

(with free variables z of type $A+B$ and x of type X). By first-order reasoning it is possible to see that the composite arrow

$$ \begin{align*}k:R\buildrel{r}\over{\hookrightarrow} (A+B)\times X \buildrel{\pi_{A+B}}\over\longrightarrow A+B \end{align*} $$

is both in ${\mathcal {M}}^\bot $ (i.e., that $\exists _k(\top )=\top $ ) and mono (i.e., that $k(z_1)=k(z_2)\longrightarrow z_1=z_2$ holds). Moreover, applying the characterization of disjoint unions in relational $\mathbf {G}$ -sets, in every conservative embedding $\eta $ of $({\mathbf {E}}, {\mathcal {M}})$ into a category of the kind $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ , $\eta (k)$ turns out to be an isomorphism. By conservativity, k is an isomorphism and then

$$ \begin{align*}A+B\buildrel{k^{-1}}\over \longrightarrow R \buildrel{r}\over {\hookrightarrow} (A+B)\times X\buildrel{\pi_X}\over \longrightarrow X \end{align*} $$

gives the required h.

(Ad (ii)). Given a zero object O and any object X, we must find a unique ${h:O\longrightarrow X}$ . Uniqueness has been observed above. For existence, consider the projection $\pi _O:O\times X \longrightarrow O$ : this is mono and becomes iso in any representation into relational $\mathbf {G}$ -sets, so it is iso by conservativity. The inverse of $\pi _O$ composed with the projection into X gives the required h.

(Ad (iii)). Suppose we are given a quotient $q:X\twoheadrightarrow Y$ . We show that q is a regular epimorphism. Take $f:X\longrightarrow Z$ with $ker(q)\leq ker(f)$ . We need to find a unique $h:Y\longrightarrow Z$ such that $h\circ q=f$ . Uniqueness comes from the fact that q is epi, as any arrow in ${\mathcal {M}}^\bot $ (recall Proposition 2.3). Consider the relation $R\buildrel {r}\over {\hookrightarrow } Y\times Z$ given by

$$ \begin{align*}\exists_{\pi_{Y\times Z}}(\pi_{X\times Z}^* (q\times 1_Y)^* {=_Y} \land \pi_{X\times Z}^* (f\times 1_Z)^* {=_Z}). \end{align*} $$

In the internal language of doctrines, this can be expressed by the formula

$$ \begin{align*}\exists x\, (q(x)= y\wedge f(x)=z). \end{align*} $$

By first-order reasoning it is possible to see that the composite arrow

$$ \begin{align*}k:R\buildrel{r}\over{\hookrightarrow} Y\times Z \buildrel{\pi_Y}\over\longrightarrow Y \end{align*} $$

is both in ${\mathcal {M}}^\bot $ and mono. Moreover, applying the characterization of quotients in relational $\mathbf {G}$ -sets, in every conservative embedding $\eta $ of $({\mathbf {E}}, {\mathcal {M}})$ into a category of the kind $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ , $\eta (k)$ turns out to be an isomorphism. By conservativity, k is an isomorphism and then

$$ \begin{align*}Y\buildrel{k^{-1}}\over \longrightarrow R \buildrel{r}\over {\hookrightarrow} Y\times Z\buildrel{\pi_X}\over \longrightarrow Z \end{align*} $$

gives the required h.

We have shown that any quotient q is the coequalizer of its kernel $ker(q)$ . Since any equivalence relation is of the form $ker(q)$ , it follows that, reciprocally, any regular epimorphism is a quotient.

(Ad (iv)). We show that the pullback of a regular epi $q: X\longrightarrow Q$ along any arrow is a regular epi. Let us factorize such a pullback $q'$ as

$$ \begin{align*}X'\buildrel{q"}\over \longrightarrow Q' \buildrel{m}\over \longrightarrow Q", \end{align*} $$

where $q"$ is a regular epi ( $q"$ is the quotient, aka the coequalizer, of the kernel of $q'$ ) and m is mono by the lemma below. In a conservative representation $\eta $ of $({\mathbf {E}}, \mathcal {M})$ into a relational $\mathbf {G}$ -sets category, it turns out $\eta (m)$ is an iso because relational $\mathbf {G}$ -sets are a regular category, hence m itself is iso by conservativity.

Lemma 6.14. In an f-regular category $({\mathbf {E}}, {\mathcal {M}})$ , suppose that $f: X\longrightarrow Y$ factors as $X\buildrel {q}\over \longrightarrow Q\buildrel {m}\over \longrightarrow Y$ , where q is such that $q\in {\mathcal {M}}^\bot $ and $ker(f)\leq ker(q)$ (this is certainly the case if q is the coequalizer of $ker(f)$ , recall Proposition 2.3). Then m is mono.

Proof. This can be established using first-order reasoning via the internal language. Alternatively, make a diagram chasing, using the fact that the class ${\mathcal {M}}^\bot $ is pullback-stable and is contained in the class of epis by Proposition 2.3.

From the above facts, it follows that in a saturated Boolean modal quasi-pretopos every arrow $f:X\longrightarrow Y$ factors as

$$ \begin{align*}X\buildrel{q}\over{\longrightarrow} X' \buildrel{\iota}\over{\longrightarrow} Y'\buildrel{m}\over{\longrightarrow} Y, \end{align*} $$

where q is a regular epi, m is a regular mono, and $\iota $ is both mono and epi; moreover such factorization is pullback stable.

7 Modal first-order theories

We present a calculus for first-order modal logic and we show how to associate a modal category with each theory so as to prove a completeness theorem for modal first-order theories. For simplicity, we supply a full classical logic calculus in Hilbert style with primitive logical operators $\rightarrow , \bot , \forall , \Box $ , where the operator $\Diamond $ for instance is defined by $\Diamond \varphi = \lnot \Box \lnot \varphi $ . Readers interested in relevant fragments (like the coherent fragment) can easily formulate by themselves the corresponding calculi using, e.g., suitable sequents.

The main problem we have to face in building a syntactic calculus corresponding to our modal categories lies in the fact that modal operators are not pullback stable, they are only ‘half-stable’ (see the continuity condition in Definition 3.1). This problem has been somewhat known for a long time in the first-order modal logic literature (especially in the literature oriented towards philosophical applications), although of course it was not formulated in this way. One of the proposed solutions in such literature was the adoption of a restricted form of $\lambda $ -abstraction [44]: this is the solution we want to imitate below, in order to obtain a calculus which is more flexible than the calculus presented in [8, 16]. The idea is to combine abstraction operators with a variant of the ‘de-modalization’ technique employed in some literature [2, 18]. The outcome of this is a full encoding of our first-order modal theories into classical first-order theories.

We need to use formulae-in-context, as commonly adopted in the categorical logic literature to handle possibly empty domains (in particular, we shall follow rather closely the formalism of [27, Section D1] when building our language and our syntactic categories). ⁵

We fix a many sorted first-order signature (with equality) $\Sigma $ : sorts are indicated with letters $X,Y, \dots $ , atomic predicates with letters $P,Q, \dots $ and function symbols with letters $f, g\,\dots $ To each predicate or function symbol an arity domain (i.e., a list of sorts) is associated; function symbols are also assigned a codomain sort. We have a countably infinite supply of variables for every sort $x_1^X, x_2^X, \dots , y_1^Y, y_2^Y, \dots $ (the sorts of the variables may be omitted if confusion does not arise). A finite list of variables without repetitions is called a context: contexts are indicated with letters $\underline {x}, \underline {y}, \dots $ (the list of the sorts of the variables in a context is often left implicit in the notation).

Terms-in-contexts are defined as follows:

1. (i) $x^X: \underline {x}$ is a term in context of type X if $x^X\in \underline {x}$ ;

2. (ii) if $t_1:\underline {x}, \dots , t_n:\underline {x}$ are terms in contexts of types $X_1,\dots , X_n$ , respectively, then $f(t_1, \dots , t_n):\underline {x}$ is a term in context of type X if the function symbol f has X as codomain sort and $X_1\ldots X_n$ as arity domain.

We now define formulae-in-context and predicates by mutual recursion.

Formulae-in-contexts are defined as follows:

1. (i) if $t_1:\underline {x}, \dots , t_n:\underline {x}$ are terms in contexts of types $X_1,\dots , X_n$ , respectively, then $P(t_1, \dots , t_n):\underline {x}$ is a formula in context, if the predicate P has $X_1\ldots X_n$ as arity domain;

2. (ii) $\bot :\underline {x}$ is a formula in context, for every context $\underline {x}$ ;

3. (iii) if $\varphi _1:\underline {x}$ and $\varphi _2:\underline {x}$ are formulae in context, so is $(\varphi _1\to \varphi _2):\underline {x}$ ;

4. (iv) if $x^X\in \underline {x}$ and $\varphi :\underline {x}$ is a formula in context, so is $(\forall x^X \varphi ): \underline {x}\setminus \{x^X\}$ ;

Predicates are so defined:

1. (a) atomic predicates P with domain arity $X_1\ldots X_n$ are predicates with domain arity $X_1\ldots X_n$ ;

2. (v) if $\varphi :\underline {x}$ (where $\underline {x}= x_1^{X_1},\dots , x_n^{X_n}$ ) is a formula in context, then

   $$ \begin{align*}\Box\, \{ \underline{x} \mid\varphi\} \end{align*} $$
   is a predicate with domain arity $X_1\ldots X_n$ .

We underline that the abstraction operator applies to the context as a whole and not to single variables. Below, if $\underline {x}= x_1^{X_1},\dots , x_n^{X_n}$ and if $t_1:\underline {y}, \dots , t_n:\underline {y}$ are terms in contexts of types $X_1,\dots , X_n$ , respectively, we may write simply

$$ \begin{align*}(\Box \varphi)(t_1, \dots, t_n) : \underline{y}\end{align*} $$

instead of $ \Box \, \{ \underline {x} \mid \varphi \} (t_1, \dots , t_n) $ . We may also write $\Box \varphi :\underline {x}$ for $(\Box \varphi )(\underline {x}):\underline {x}$ .

Remark 7.1. This special way of treating the connective $\Box $ is well known in the type theory of modal logic, where it is called delayed substitution [5, 24]. Since $\Box $ does not commute with substitution, we cannot use the straightforward recursive definition to substitute terms for free variables. Instead, substitutions accumulate just outside of formulas of the form $\Box \varphi $ without possible simplification, hence the need of formulas of the form $(\Box \varphi )(t_1,\dots ,t_n)$ .

We treat our formulae modulo $\alpha $ -conversion, meaning that alphabetic variants will be considered the same formula (notice that both $\forall $ and $\{ \underline {x} \mid -\}$ are variable binders). Substitution can be inductively defined as expected: if $y_1^{Y_1}\ldots y_m^{Y_m}$ are the variables of the context $\underline {y}$ of a formula in context $\varphi :\underline {y}$ and if $u_1:\underline {x}, \dots , u_m:\underline {x}$ are terms in context of types $Y_1,\dots , Y_m$ , respectively, we define the formula in context

$$ \begin{align*}\varphi[u_1/y_1^{Y_1}, \dots, u_m/y_m^{Y_m}]: \underline{x} \end{align*} $$

as follows (we assume that the bounded variables occurring in $\varphi $ are disjoint from $\underline {x}$ —if it is not so, an alphabetic variant of $\varphi $ is taken before applying the definition below):

1. (i) for every predicate P (atomic or not), we define

   $$ \begin{align*}P(t_1, \dots, t_n)[u_1/y_1^{Y_1}, \dots, u_m/y_m^{Y_m}]:\underline{x}\end{align*} $$
   as the formula in context $P(t^{\prime }_1, \dots , t^{\prime }_n):\underline {x}$ , where $t^{\prime }_i$ (for $1\leq i\leq n$ ) is obtained from $t_i$ by replacing in it each occurrence of $y_1^{Y_1}, \dots , y_m^{Y_m}$ by $u_1, \dots , u_m$ , respectively;

2. (ii) $\bot [u_1/y_1^{Y_1}, \dots , u_m/y_m^{Y_m}]:\underline {x}$ is the formula in context $\bot :\underline {x}$ ;

3. (iii) $(\varphi _1\to \varphi _2)[u_1/y_1^{Y_1}, \dots , u_m/y_m^{Y_m}]:\underline {x}$ is the formula in context given by $(\varphi _1[u_1/y_1^{Y_1}, \dots , u_m/y_m^{Y_m}]\to \varphi _1[u_1/y_1^{Y_1}, \dots , u_m/y_m^{Y_m}]):\underline {x}$ ;

4. (iv) $(\forall x^X \varphi ) [u_1/y_1^{Y_1}, \dots , u_m/y_m^{Y_m}]: \underline {x}$ is the formula in context given by $\forall x^X \varphi [u_1/y_1^{Y_1}, \dots , u_m/y_m^{Y_m}, x^X/x^X]: \underline {x}$ .

Below we shall usually write $ \varphi [u_1/y_1^{Y_1}, \dots , u_m/y_m^{Y_m}]: \underline {x} $ as

$$ \begin{align*}\varphi[u_1, \dots, u_m]: \underline{x} \end{align*} $$

(or just as $\varphi [\underline {u}]$ if $\underline {u}=u_1, \dots , u_m$ ) for simplicity, if confusion does not arise. As a special case, for a predicate P (atomic or not), we may use the notations $P(t_1, \dots , t_n):\underline {x}$ and $P[t_1, \dots , t_n]:\underline {x}$ interchangeably. Finally, $ \varphi [u_1/y_1^{Y_1}, \dots , u_m/y_m^{Y_m}]: \underline {x} $ can be abbreviated as $\varphi [u_i/y_i^{Y_i}]$ if $u_j$ is equal to $y_j$ for all $j\neq i$ .

Axioms and rules for the calculus are specified in Table 1. Few observations are in order: $(Taut)$ , $(\Box Dis)$ , $(MP)$ , and $(Nec)$ are the standard axiomatization of the modal system K, as reported in modal logic textbook like [9]. On the other hand, $(\forall $ - $Ex)$ , $(\forall $ - $In),(Refl)$ , and $(Rep)$ are the well-known Hilbert-style axiomatization of classical first-order logic with equality. The only specific axiom of our framework in the continuity axiom $(Cont)$ . Rule $(Inst)$ is admissible in the pure calculus and is reported here in analogy to [27, D1].

Table 1 The first-order modal calculus.

Given a modal theory T (i.e., a set of formulae in context) and a formula in context $\varphi :\underline {x}$ , we write

$$ \begin{align*}\vdash^{\underline{x}}_T \varphi \end{align*} $$

(and we say that $\varphi :\underline {x}$ is derivable from T) to mean that there is a derivation of $\varphi :\underline {x}$ that uses the above axioms and rules together with the formulae of T as extra axioms. Notations like $\varphi _1, \dots , \varphi _n \vdash ^{\underline {x}}_T \varphi $ mean $\vdash ^{\underline {x}}_T \varphi _1\wedge \cdots \wedge \varphi _n \rightarrow \varphi $ .

An evaluation of the signature $\Sigma $ into a Boolean modal category $({\mathbf {E}}, \mathcal {M})$ is a map $\mathcal {I}$ that associates

• • with every sort X of $\Sigma $ an object $\mathcal {I}(X)$ of ${\mathbf {E}}$ ;

• • with every atomic predicate P of $\Sigma $ of domain $X_1\ldots X_n$ an $ \mathcal {M}$ -subobject $\mathcal {I}(P)$ of $\mathcal {I}(X_1)\times \cdots \times \mathcal {I}(X_n)$ ;

• • with every function symbol f of $\Sigma $ of arity domain $X_1\ldots X_n$ and codomain X an arrow $\mathcal {I}(f): \mathcal {I}(X_1)\times \cdots \mathcal {I}(X_n)\longrightarrow \mathcal {I}(X)$ in ${\mathbf {E}}$ .

By recursion, it is clear how to extend $\mathcal I$ to terms and formulae in context, so as to associate with a formula in context $\varphi :\underline {x}$ (where $\underline {x}=x_1^{X_1},\dots , x_n^{X_n}$ ) an $ \mathcal {M}$ -subobject $\mathcal {I}(\varphi )\hookrightarrow \mathcal {I}(X_1)\times \cdots \mathcal {I}(X_n)$ . The evaluation is called a model of a modal theory T in $({\mathbf {E}}, \mathcal {M})$ iff for every $\varphi :\underline {x}$ in T we have that $\top =\mathcal {I}(\varphi )$ .

The following categorical completeness theorem links our calculus with modal categories.

Theorem 7.2. We have $\vdash _T^{\underline {x}} \varphi $ iff $\top =\mathcal {I}(\varphi )$ holds for every model $\mathcal I$ of T in a Boolean modal category $({\mathbf {E}}, \mathcal {M})$ .

Proof. One side of the theorem below is just an induction on the length of derivations, whereas the other side requires the construction of a modal category $({\mathbf {E}}_T, \mathcal {M}_T)$ associated with a modal theory T. We shall merely sketch here such a construction, relying on the formalism taken from [27]. Although the detailed computations involved in the construction are indeed rather laborious, the construction by itself is interesting because it shows how to build the suitable factorization system, taking inspiration from the conditions of Definition 3.1.

We take as objects the equivalence classes of formulae in context

$$ \begin{align*}\{ \varphi: \underline{x}\}, \end{align*} $$

where we consider $\varphi :\underline {x}$ and $\varphi ':\underline {x}'$ to be equivalent if $\varphi ':\underline {x}'$ is a renaming of $\varphi :\underline {x}$ (this means that there is an alphabetic variant $\tilde \varphi :\underline {x}$ of $\varphi :\underline {x}$ and a variable bijection $\underline {x}\longmapsto \underline {x}'$ such that $\varphi ':\underline {x}'$ is equal to $\tilde \varphi [\underline {x}'/\underline {x}] : \underline {x}'$ ). In defining morphisms from $\{ \varphi : \underline {x}\}$ to $\{ \psi : \underline {y}\}$ we may assume that the contexts $\underline {x}$ and $\underline {y}$ are disjoint. A formula in context $R:\underline {x},\underline {y}$ such that

$$ \begin{align*} & R\vdash^{\underline{x},\underline{y}}_T \varphi\wedge \psi \\ & R[\underline{x}, \underline{y}], R[\underline{x}, \underline{y}']\vdash^{\underline{x},\underline{y},\underline{y}'}_T \underline{y}=\underline{y}' \\ & \varphi\vdash^{\underline{x}}_T \exists \underline{y} R \end{align*} $$

is called a functional relation between $\varphi : \underline {x}$ and $\psi : \underline {y}$ (here $\underline {y}=\underline {y}'$ means the componentwise conjunction of equalities). We take as arrows between $\{ \varphi : \underline {x}\}$ and $\{ \psi : \underline {y}\}$ the equivalence classes (wrt to provable equivalence in T) of such functional relations that satisfy in addition the following requirement for all formulae in context $\theta : \underline {y} ,\underline {w}$ with $\underline {w}\cap \underline {x}=\emptyset $ :

(28) $$ \begin{align} \varphi[\underline{x}]\wedge \Diamond \exists \underline{y} (R[\underline{x},\underline{y}]\wedge \theta[\underline{y},\underline{w}])\vdash^{\underline{x},\underline{w}}_T \exists \underline{y} (R[\underline{x},\underline{y}]\wedge (\Diamond \theta)[\underline{y},\underline{w}]). \end{align} $$

This formula is the transcription in our language of the continuity condition (20) for a relation R in a modal category (the extra conjunction with $\varphi [\underline {x}]$ appearing in (28) is due to the fact that $\varphi $ itself is meant to be a subobject of the object represented by the context $\underline {x}$ ).

Following [27], the composite of two morphisms

is $\{\exists \underline {y} (R[\underline {x},\underline {y}]\wedge S[\underline {y},\underline {z}])\}$ (notice that the context of the formulae representing an arrow can be desumed from the domain and the codomain), whereas the identity of $\{\varphi :\underline {x}\}$ is

Associativity and identity conditions can be checked as in [27], but the continuity condition (28) has to be checked in addition (we omit the sufficiently straightforward syntactic details).

Removing the continuity condition (28) and repeating the considerations in [27], we obtain a coherent Boolean category (that we call $\mathbf {C}_T$ ) and an obvious faithful inclusion functor ${\mathbf {E}}_T \longrightarrow \mathbf {C}_T$ : the first fact to check is that this functor preserves finite limits.

In $\mathbf {C}_T$ , the terminal object is $\{\top :\; \}$ (its context is empty) and $\{ \varphi : \underline {x}\}\buildrel {\{\varphi \}}\over \longrightarrow \{\top :\; \}$ is the unique map from $\{ \varphi : \underline {x}\}$ to the terminal object. The product of $\{ \varphi : \underline {x}\}$ and $\{ \psi : \underline {y}\}$ is $\{ \varphi [\underline {x}]\wedge \psi [\underline {y}]: \underline {x},\underline {y}\}$ and the projection maps are

The equalizer of $\{\varphi :\underline {x}\}\buildrel {\{R\}}\over \longrightarrow \{\psi :\underline {y}\}$ and of $\{\varphi :\underline {x}\}\buildrel {\{S\}}\over \longrightarrow \{\psi :\underline {y}\}$ is

Since all involved maps (including the ones needed for the universal properties of limits) satisfy the continuity condition (28), we have that indeed ${\mathbf {E}}_T$ is a lex subcategory of $\mathbf {C}_T$ .

We now have to identify an adequate stable factorization system in $ {\mathbf {E}}_T$ . In the end, we shall see that the above faithful inclusion preserves the chosen factorization systems (which is the extremal epi/mono factorization system in $\mathbf {C}_T$ ), but some subtleties arise, due to the fact that iso’s are not reflected (in fact, an iso in $\mathbf {C}_T$ is iso in ${\mathbf {E}}_T$ too iff both it and its inverse satisfy the continuity condition (28)).

We take as the right class $\mathcal {E}_T$ in $\mathbf {C}_T$ the set of the maps $\{ \varphi : \underline {x}\}\buildrel {\{R\}}\over \longrightarrow \{ \psi : \underline {y}\}$ satisfying the condition $\psi \vdash ^{\underline {y}} \exists \underline {x}\, \varphi $ . These are the maps that are strong epi in $\mathbf {C}_T$ . Now, like in $\mathbf {C}_T$ , every map factors in ${\mathbf {E}}_T$ as a map in $\mathcal {E}_T$ followed by a ‘canonical subobject’, i.e., by a map of the kind

where $\varphi \vdash _T^{\underline {x}} \psi $ . Here comes however the main difference: in $\mathbf {C}_T$ , every monic arrow is isomorphic to a canonical subobject (see [27] for a proof), whereas this is not true anymore in ${\mathbf {E}}_T$ because we lack sufficiently many isomorphisms. This means that the closure under isomorphisms of the class of canonical subobjects in ${\mathbf {E}}_T$ is a more restricted class (still orthogonal to $\mathcal {E}_T$ , though), which we are going to identify.

Recall [27] that an arrow $\{ \varphi : \underline {x}\}\buildrel {\{M\}}\over \longrightarrow \{ \psi : \underline {y}\}$ is monic (in $\mathbf {C}_T$ and also in ${\mathbf {E}}_T$ ) iff its satisfies the condition

$$ \begin{align*}M(\underline{x}_1,\underline{y}), M(\underline{x}_2,\underline{y}) \vdash^{\underline{x}_1, \underline{x}_2,\underline{y}} \underline{x}_1 = \underline{x}_2. \end{align*} $$

We let $ \mathcal {M}_T$ be the class of monic arrows of ${\mathbf {E}}_T$ satisfying the further condition (below we use notations like $ \zeta _1=^{\underline {z}}_T \zeta _2$ to mean ‘ $\zeta _1\vdash _T^{\underline {z}} \zeta _2$ and $\zeta _2\vdash _T^{\underline {z}} \zeta _1$ ’)

(29) $$ \begin{align} \varphi[\underline{x}] \wedge \Diamond \theta~ =^{\underline{x},\underline{w}}_T ~\exists \underline{y}\,(M[\underline{x},\underline{y}]\wedge (\Diamond \exists \underline{x}' ( M[\underline{x}',\underline{y}] \wedge \theta[\underline{x}',\underline{w}]))[\underline{y},\underline{w}]) \end{align} $$

for every formula $\theta :\underline {x},\underline {w}$ such that $\theta \vdash ^{\underline {x},\underline {w}}_T \varphi [\underline {x}]$ . Notice that (29) is nothing but a ‘translation’ of condition (19).

The following facts can now be checked via suitable syntactic proofs:

1. (i) $ \mathcal {M}_T$ is orthogonal to $\mathcal {E}_T$ ;

2. (ii) $ \mathcal {M}_T$ is closed under composition and contains iso’s;

3. (iii) if $\{M\}, \{M'\}$ are in $ \mathcal {M}_T$ and $\{ M\}=\{ M'\}\circ \{R\}$ , then $\{R\}$ is in $ \mathcal {M}_T$ too.

   

As a consequence we have that every arrow $\{M\}$ in $ \mathcal {M}_T$ is isomorphic to a canonical subobject: this is because the first component of the factorization of $\{M\}$ (as an arrow in $\mathcal {E}_T$ followed by a canonical subobject) is in $\mathcal {E}_T\cap \mathcal {M}_T$ , so it is an iso.

To complete the construction of $({\mathbf {E}}_T, \mathcal {M}_T)$ , we need to introduce the modal operators. This is done as follows: if $\{ M\}\simeq \{\psi [\underline {x}'/\underline {x}]:\underline {x}'\}$ is an $ \mathcal {M}_T$ -subobject of $\{\varphi : \underline {x}\}$ we let

$$ \begin{align*}\Diamond \{M\}~:=~\{(\varphi \wedge \Diamond \psi)[\underline{x}'/\underline{x}]:\underline{x}'\}. \end{align*} $$

The conditions of Definition 3.1 are easily seen to be satisfied by construction.

It is now straightforward to define a canonical evaluation $\mathcal {I}_T$ such that we have $\mathcal {I}_T(\varphi :\underline {x})=\{\varphi :\underline {x}\}$ for every formula $\varphi :\underline {x}$ . The claim of the theorem follows immediately.

8 Conclusion and further developments

We proposed an approach to quantified modal logic via logical categories: our modal categories are lex categories endowed with an adequate factorization system $({\mathcal {E}}, \mathcal {M})$ , where the semilattices of $ \mathcal {M}$ -subobject are distributive lattices with a modal operator satisfying the axioms of modal logic K. These modal distributive lattices interact with the factorization system via continuity conditions for inverse images and subspace conditions for $ \mathcal {M}$ -subobjects. The interaction becomes tighter when (mutually equivalent) saturation axioms for $ \mathcal {M}$ -subobjects, definable functional relations and definable isomorphisms are assumed. To justify our framework we proved representation theorems in semantic modal categories which are quasi-toposes and realize a counterpart semantics. Still, we believe that our understanding of modal categories needs more investigation: in this section, we mention some problems and some directions for future research.

1. 1. The relationship between modal categories and modal first-order theories needs to be studied further. The reader might have noticed that the modal category $({\mathbf {E}}_T, \mathcal {M}_T)$ built in Section 7 is a saturated modal category. Should this fact give strong motivation towards the adoption of the saturation principles in the very definition of a modal category? Not quite. The saturation principles are essentially higher order principles: they analyze conditions under which a definable subobject/functional relation/bijection is promoted to become a real subspace/continuous function/isomorphism. The conditions are relative to the available $\mathcal {M}$ -subobjects (we would in fact say: “to the available current language”), so they might fail in a larger environment where more $ \mathcal {M}$ -subobjects arise. Thus, there is no reason why the internal models of a modal first-order theory T into a modal category $({\mathbf {E}}, \mathcal {M})$ should correspond to modal functors from $({\mathbf {E}}_T, \mathcal {M}_T)$ to $({\mathbf {E}}, \mathcal {M})$ . Such property is likely to be enjoyed by a smaller modal category than $({\mathbf {E}}_T, \mathcal {M}_T)$ , namely the ‘minimum modal subcategory’ of $({\mathbf {E}}_T, \mathcal {M}_T)$ comprising the interpretation of terms and of predicates of the language of T as arrows and as $ \mathcal {M}$ -subobjects, respectively. Such a smaller category will have less isomorphisms, so the closure under isomorphisms of the ‘canonical subobjects’ class will be different than in $({\mathbf {E}}_T, \mathcal {M}_T)$ . Thus, modal categories might carry more information than modal first-order theories, in the sense that they also declare explicitly that certain subobjects/functional relations/bijections have to be considered as real subspaces/continuous functions/isomorphisms. Such extra information is partially undetermined in a modal first-order theory and it can be maximized or minimized going though appropriate modal subcategories of $({\mathbf {E}}_T, \mathcal {M}_T)$ ; for a minimal choice in this sense, see the construction of a syntactic modal category out of a modal Lawvere doctrine in [19, Section 5].

2. 2. Adding $S4$ axioms (7) is not as harmless as one may expect. From the semantic side, $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ should be replaced by a relational presheaf category $\mathbf {Rel^C}$ [8, 15, 38]: here $\mathbf {C}$ is a category (not just a graph), objects are lax set-valued functors and arrows are the same as in $\mathbf {Rel^C}$ . This category is not as well-behaved as $\operatorname {\mathrm {\mathbf {Rel}}}^{\mathbf {G}}$ (to see it, notice that $\mathbf {Rel^C}$ is $\mathbf {POr}$ , when $\mathbf {C}$ is the singleton category, so it is not a quasi-topos). A completeness theorem for modal first-order theories has been proved in [8, 15], but the extension to modal categories does not look easy. The reason is that the preservation of the $ \mathcal {M}$ class is responsible for the introduction of the reflection condition (10) in the notion of a modal transformation and this reflection condition is not directly compatible with the lax composability that modal transformations should satisfy in the new $S4$ environment. Once again, these extra difficulties show that moving from modal first-order theories to the more expressive framework of modal categories is far from trivial.

3. 3. Obtaining sufficient conditions for a representation theorem into a (power of) $\mathbf {Top}$ is more challenging, but it has been achieved in [19]. It was shown in [16] that the conditions (7) and (8) are sufficient to prove a completeness theorem for a first-order modal theory in a purely relational signature. In order to move from modal first-order theories to modal categories and deal with functional symbols and embeddings, one has to add a further ‘loop contraction principle’ (see [19]).

4. 4. The mechanism for introducing modalities of the present article does not require full first-order logic, it operates already in the fragment of coherent logic. However, it is possible to weaken this fragment further. In fact we need existential quantifiers only in order to formulate the subspace condition (6), but a closer inspection of this condition reveals that it requires only direct images along monos, so condition (6) makes sense in the essentially algebraic fragment of lex categories. Disjunctions are needed to formulate K-axioms, but K-axioms might be weakened themselves. This is important in the description logics framework, where lighter logics are introduced. In particular, the description logic $\mathcal {EL}$ requires from the algebraic side just $\wedge $ -semilattices with a monotone operator $\Diamond $ . $\mathcal {EL}$ plays an important role in the applications because it is computationally tractable and relevant industrial size ontologies can be formalized in $\mathcal {EL}$ . It is evident that our framework offers a natural candidate for the formalization of a predicate extension of $\mathcal {EL}$ in the purely essentially algebraic context.

5. 5. As an opposite research direction, one should explore extensions to higher order modal logic: the fact that our main semantic frameworks (namely relational $\mathbf {G}$ -sets) are quasi-toposes makes this perspective viable.