Related papers: A simplicial category for higher correspondences
In this thesis we define the notion of a locally stratified space. Locally stratified spaces are particular kinds of streams and d-spaces which are locally modelled on stratified spaces. We construct a locally presentable and cartesian…
We formulate a model-independent theory of co/cartesian morphisms and co/cartesian fibrations: that is, one which resides entirely *within the $\infty$-category of $\infty$-categories*. We prove this is suitably compatible with the…
We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing "profinite" analogues of known model categories. Our construction quickly recovers Morel's…
Theorem (after Giraud, SGA 4): Suppose $A$ is a simplicial category. The following conditions are equivalent: (i) There is a cofibrantly generated closed model category $M$ such that $A$ is equivalent to the Dwyer-Kan simplicial…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial…
A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets…
In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete…
We study the interaction between the notions of filteredness, fractions and fibrations in the theory of bicategories, generalizing classical results for categories. We give an explicit formula for filtered pseudo-colimits of categories…
We explain how homotopical information of two composeable relations can be organized in two simplicial categories that augment the relations row and column complexes. We show that both of these categories realize to weakly equivalent…
nspired by the work of J$\o$rgensen [J], we define a (upper-, lower-) symmetric recollements; and give a one-one correspondence between the equivalent classes of the upper-symmetric recollements and one of the lower-symmetric recollements,…
We introduce a new higher categorical structure called a weakly globular n-fold category. This structure is based on iterated internal categories and on the notion of weak globularity. We identify a suitable class of pseudo-functors whose…
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 Lagrangian correspondences between Liouville manifolds and construct functors between wrapped Fukaya categories. The study naturally brings up the question on comparing two versions of wrapped Fukaya categories of the product…
This paper defines double fibrations (fibrations of double categories) and describes their key examples and properties. In particular, it shows how double fibrations relate to existing fibrational notions such as monoidal fibrations and…
This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with…
We give a definition of a coherent adjunction in a $4$-category consisting of a finite list of $k$-morphisms for $k\leq 4$, plus equations beetween $4$-morphisms. We prove that the restriction map from the space of coherent adjunctions in a…
In this paper, we show that two constructions form stacks: Firstly, as one varies the $\infty$-topos, $\mathcal{X}$, Lurie's homotopy theory of higher categories internal to $\mathcal{X}$ varies in such a way as to form a stack over the…
The aim of these notes is to introduce the intuition motivating the notion of a "complicial set", a simplicial set with certain marked "thin" simplices that witness a composition relation between the simplices on their boundary. By varying…
We upgrade the classical operation of \textit{isomonodromic deformations} along a path $\gamma$ to a functor $\mathbb{P}_{\gamma}$ between categories of flat connections with logarithmic singularities along a divisor $D$, which itself…