For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X),BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization |NB| of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties.
It is known that a topological correspondence \((X,λ)\) from a locally compact groupoid with a Haar system \((G,α)\) to another one, \((H,β)\), produces a \(\textrm{C}^*\)-correspondence \(\mathcal{H}(X,λ)\) from \(\textrm{C}^*(G,α)\) to \(\textrm{C}^*(H,β)\). In one of our earlier article we described composition two topological correspondences. In the present article, we prove that second countable locally compact Hausdorff topological groupoids with Haar systems form a bicategory \(\mathfrak{T}\) when equipped with a topological correspondences as 1-arrows. The equivariant homeomorphisms of…
The homotopy theory of higher categorical structures has become a relevant part of the machinery of algebraic topology and algebraic K-theory, and this paper contains contributions to the study of the relationship between Bénabou's bicategories and the homotopy types of their classifying spaces. Mainly, we state and prove an extension of Quillen's Theorem B by showing, under reasonable necessary conditions, a bicategory-theoretical interpretation of the homotopy-fibre product of the continuous maps induced on classifying spaces by a diagram of bicategories $\mathcal{A}\to\mathcal{B}\leftarrow …
For a topos $\mathbf{T}$, there is a bicategory $\mathbf{MonicSp(Csp(T))}$ whose objects are those of $\mathbf{T}$, morphisms are cospans in $\mathbf{T}$, and 2-morphisms are isomorphism classes of monic spans of cospans in $\mathbf{T}$. Using a result of Shulman, we prove that $\mathbf{MonicSp(Csp(T))}$ is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as $2$-morphisms inside a compact closed sub-bicategory of $\mathbf{MonicSp(Csp(Graph))}$.
Reasoning in the 2-category Con of contexts, certain sketches for arithmetic universes (i.e. list arithmetic pretoposes; AUs), is shown to give rise to base-independent results of Grothendieck toposes, provided the base elementary topos has a natural numbers object. Categories of strict models of contexts $T$ in AUs are acted on strictly on the left by non-strict AU-functors and strictly on the right by context maps, and the actions combine in a strict action of a Gray tensor product. Any context extension $T_0 \subset T_1$ gives rise to a bundle. For each point of $T_0$ - a model $M$ of $T_0$…
Every principal G-bundle is classified up to equivalence by a homotopy class of maps into the classifying space of G. On the other hand, for every nice topological space Milnor constructed a strict model of loop space, that is a group. Moreover the morphisms of topological groups defined on the loop space of X generate all the bundles over X up to equivalence. In this paper, we show that the relationship between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to …
In this paper we present classifying toposes for the following theories: the theory of $\mathcal{C}^{\infty}-$rings, the theory of local $\mathcal{C}^{\infty}-$rings and the theory of von Neumann regular $\mathcal{C}^{\infty}-$rings. The classifying toposes for the first two theories were stated without proof by Ieke Moerdijk and Gonzalo Reyes on the page 366 of \cite{MSIA}, where they assert that the topos ${\rm \bf Set}^{\mathcal{C}^{\infty}{\rm \bf Rng}_{\rm fp}}$ classifies the theory of $\mathcal{C}^{\infty}-$rings and that the smooth Zariski topos classifies the theory of local $\mathcal…
For a collection of subcategories satisfying a fixed set of conditions, for example thick subcategories of a triangulated category, we define a topological space called classifying space of subcategories. We show that this space classifies various prime subcategories in the sense that they bijectively correspond to the closed subsets of the classifying space. Many well-known results of subcategory classification fit into this framework. An example which cannot be classified by a topological space in the above sense is also given.
Given a regular cardinal $κ$ such that $κ^{<κ}=κ$, we study a class of toposes with enough points, the $κ$-separable toposes. These are equivalent to sheaf toposes over a site with $κ$-small limits that has at most $κ$ many objects and morphisms, the (basis for the) topology being generated by at most $κ$ many covering families, and that satisfy a further exactness property $T$. We prove that these toposes have enough $κ$-points, that is, points whose inverse image preserve all $κ$-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $κ=ω$, …
