相关论文: The groupoidal analogue Theta~ to Joyal's category…

We prove that for any test category $A$, in the sense of Grothendieck, satisfying a compatibility condition between homology equivalences and weak equivalences of presheaves, the homotopy category of abelian presheaves on $A$ is equivalent…

We introduce the notion of groupoidal (weak) test category, which is a small category A such that the groupoid-valued presheaves over A models homotopy types in a "canonical and nice" way. The definition does not require a priori that A is…

This work originates from chapters V and VII of Grothendieck's manuscript Pursuing Stacks, which contains a series of questions, as well as a previously unexplored formalism, concerning the interactions between the notion of test categories…

We introduce a notion of $\Theta$-categories, which is a refinement of the notion of symmetric monoidal $\infty$-categories. We use this notion to prove a Tannakian duality statement, relating $\Theta$-categories with fpqc-stacks by means…

There is a well-established homotopy theory of simplicial objects in a Grothendieck topos, and folklore says that the weak equivalences are axiomatisable in the geometric fragment of $L_{\omega_1, \omega}$. We show that it is in fact a…

We explore the canonical Grothendieck topology and a new homotopical analog. First we discuss some background information, including defining a new 2-category called the Index-Functor Category and a sieve generalization. Then we discuss a…

We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are the translation to the world of $n$-quasi-categories of groupoidal and truncated $(\infty, n)$-$\Theta$-spaces defined by Rezk. We show that these objects are the…

We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial…

Given a small simplicial category $\C$ whose underlying ordinary category is equipped with a Grothendieck topology $\tau$, we construct a model structure on the category of simplicially enriched presheaves on $\C$ where the weak…

We show that the category of presheaves of simplicial groupoids on a site $\mathcal{C}$ is a right proper simplicial model category. We define $G$-torsor of presheaf of 2-groupoids $G$, presheaf of simplicial groups $G$ and presheaf of…

We construct a "diagonal" cofibrantly generated model structre on the category of simplicial objects in the category of topological categories sCat_{Top}, which is the category of diagrams [\Delta^{op}, Cat_{Top}]. Moreover, we prove that…

A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the…

We prove that every Grothendieck topology induces a hereditary torsion pair in the category of presheaves of modules on a ringed site, and obtain a homological characterization of sheaves of modules: a presheaf of modules is a sheaf of…

We introduce a dependent type theory whose models are weak {\omega}-categories, generalizing Brunerie's definition of {\omega}-groupoids. Our type theory is based on the definition of {\omega}-categories given by Maltsiniotis, himself…

We give a new proof that the opposite of Joyal's disk category $\mathcal{D}_n$ is Berger's wreath product category $\Theta_n = \Delta\wr\cdots\wr\Delta$. Our techniques continue to apply when the simplex category $\Delta$ is replaced by…

We define a notion of groupoidal 2-quasi-categories and show that they are the fibrant objects of a model structure on the category of $\Theta_2$-sets. We show that this model category is Quillen equivalent to the Kan-Quillen model category…

We prove that the category of trees $\Omega$ is a test category in the sense of Grothendieck. This implies that the category of dendroidal sets is endowed with the structure of a model category Quillen-equivalent to spaces. We show that…

The goal of this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the $\infty$-categorical approach, as developed by Lurie. Three…

We study (vertically) normal lax double functors valued in the weak double category $\mathbb{C}\mathrm{at}$ of small categories, functors, profunctors and natural transformations, which we refer to as lax double presheaves. We show that for…

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper…