We develop realizability models of intensional type theory, based on groupoids, wherein realizers themselves carry non-trivial (non-discrete) homotopical structure. In the spirit of realizability, this is intended to formalize a homotopical BHK interpretation, whereby evidence for an identification is a path. Specifically, we study partitioned groupoidal assemblies. Categories of such are parameterized by “realizer categories” (instead of the usual partial combinatory algebras) that come equipped with an interval qua internal cogroupoid. The interval furnishes a notion of homotopy as well as a fundamental groupoid construction. Objects in a base groupoid are realized by points in the fundamental groupoid of some object from the realizer category; isomorphisms in the base groupoid are realized by paths in said fundamental groupoid. The main result is that, under mild conditions on the realizer category, the ensuing category of partitioned groupoidal assemblies models intensional (1-truncated) type theory without function extensionality. Moreover, when the underlying realizer category is “untyped,” there exists an impredicative universe of 1-types (the modest fibrations). This is a groupoidal analog of the traditional situation.

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$-algebra generated by a semigroup of isometries $\{v_{(m,n)}\,:\, m,n \in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c\,:\,\mathbb{Z}^{2} \to \mathbb{R}$. In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$, we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.

Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo’s original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture.

My goal is to give an accessible introduction to Martin’s work on the groupoid model and how it is related to the recent notion of univalence in Homotopy Type Theory while sharing some memories of Martin.

A one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

We present definitions of homology groups Hn (p), n ≥ 0, associated to a complete type p. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H2(p) for strong types in stable theories and show that any profinite abelian group can occur as the group H2 (p).

In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid ${{C}^{*}}$-algebras. We show that the ${{C}^{*}}$-algebra $C\left( S_{q}^{2\ell +1} \right)$ is generated by an inverse semigroup $T$ of partial isometries. We show that the groupoid ${{\mathcal{G}}_{tight}}$ associated with the inverse semigroup $T$ by Exel is exactly the same as the groupoid considered by Sheu.

In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid , this defines a periodic cohomology theory on the category of finite -CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.

Adapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.

AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

In [7], a notion of noncommutative tangent space is associated with a conical pseudomanifold and Poincaré duality in K-theory is proved between this space and the pseudomanifold. The present paper continues this line of work. We show that an appropriate presentation of the notion of symbol on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret Poincaré duality in the singular setting as a noncommutative symbol map.

We show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced $C^\ast$-algebra.

In this paper we study a Haagerup inequality in the general case of discrete groupoids. We develop two geometrical tools, pinching and tetrahedral change of faces, based on deformation of triangles, to prove it. We show how to use these tools to find all the already known results just by manipulating triangles. We use these tools for groups acting freely and by isometries on the set of vertices of any affine building and give a first reduction of this inequality to its verification on some special triangles and prove the inequality when the building is of type $\tilde{A}_{k_1}\times\cdots\times\tilde{A}_{k_n}$, where $k_i\in\{1,2\}$, $i=1,\dots,n$.

Lie algebroids cannot always be integrated into Lie groupoids. We introduce a new structure, ‘Weinstein groupoid’, which may be viewed as stacky groupoids. We use this structure to present a solution to the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into such a groupoid.

Let $Y \rightrightarrows X$ be a finite flat groupoid scheme with $X$ a quasi-projective variety and let $S$ be its coarse moduli scheme. We associate to the groupoid scheme a coherent sheaf of algebras $\mathcal{O}_{X / Y}$ on $S$ which we call the non-commutative coordinate ring of the groupoid scheme. We show that when $X$ is a smooth curve and the groupoid action is generically free, the non-commutative coordinate rings which can occur are, up to Morita equivalence, the hereditary orders on smooth curves. This gives a bijective correspondence between smooth one-dimensional Deligne–Mumford stacks of finite type and Morita equivalence classes of hereditary orders on smooth curves.

The purpose of this paper is to give a detailed study of the basic theory of C*-categories. The study includes some examples of C*-categories that occur naturally in geometric applications, such as groupoid C*-categories, and C*-categories associated to structures in coarse geometry. We conclude the paper with a brief survey of Hilbert modules over C*-categories.

2000 Mathematical Subject Classification: 18D99, 46L05, 46L08.