Related papers: On cofinal functors of $\infty$-bicategories

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets, on bisimplicial spaces, on bisimplicial sets, on marked simplicial spaces. The main…

Let $X^\bullet$ be a cosimplicial object in a pointed $\infty$-category. We show that the fiber of $\mathrm{Tot}_m(X^\bullet) \to \mathrm{Tot}_n(X^\bullet)$ depends only on the pointed cosimplicial object $\Omega^k X^\bullet$ and is in…

We fix any pair $(\mathbf{\mathscr{C}},\mathbf{W})$ consisting of a bicategory and a class of morphisms in it, admitting a bicalculus of fractions, i.e. a "localization" of $\mathbf{\mathscr{C}}$ with respect to the class $\mathbf{W}$. In…

We consider the 3-category $2\mathfrak{C}at$ whose objects are 2-categories, 1-morphisms are lax functors, 2-morphisms are lax transformations and 3-morphisms are modifications. The aim is to show that it carries interesting…

In this paper we introduce and investigate the notion of semiseparable functor. One of its first features is that it allows a novel description of separable and naturally full functors in terms of faithful and full functors, respectively.…

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphism and…

We introduce a candidate for the inner hom for $Dbl^{st}_{lx}$, the category of strict double categories and lax double functors, and characterize a lax double functor into it obtaining a lax double quasi-functor. The latter consists of a…

For a strongly connected category $\mathcal C$ with pair-wise coproducts, we introduce a cosimplicial object, which serves as a sort of resolution for computing higher derived functors of ${\sf lim} : \mathrm{Ab}^{\mathcal C}\to…

The notion of the center of an algebra over a field k has a far reaching generalization to algebras in monoidal categories. The center then lives in the monoidal center of the original category. This generalization plays an important role…

In this thesis, we develop the theory of bifibrations of polycategories. We start by studying how to express certain categorical structures as universal properties by generalising the shape of morphism. We call this phenomenon…

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we…

Assume $k$ is a field and let $F:C\rightarrow Vect_{k}$ be a small $k$-linear functor from a $k$-linear abelian category $C$ to the category of vector spaces over the field $k$, the purpose of this note is to use a little knowledge of…

Abstract inner automorphisms can be used to promote any category into a 2-category, and we study two-dimensional limits and colimits in the resulting 2-categories. Existing connected colimits and limits in the starting category become…

We fix any bicategory $\mathscr{A}$ together with a class of morphisms $\mathbf{W}_{\mathscr{A}}$, such that there is a bicategory of fractions $\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]$. Given another such pair…

We reconcile the two different category-theoretic semantics of regular theories in predicate logic. A 2-category of `regular fibrations' is constructed, as well as a 2-category of `regular proarrow equipments', and it is shown that the two…

In an abelian category, the (bi)fibration of subobjects is isomorphic to the (bi)fibration of quotients. This property captures substantial information about the exactness structure of a category. Indeed, as it was shown by the second…

We apply the theory of weighted bicategorical colimits to study the problem of existence and computation of such colimits of birepresentations of finitary bicategories. The main application of our results is the complete classification of…

By homotopy linear algebra we mean the study of linear functors between slices of the $\infty$-category of $\infty$-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices…

We study lax epimorphisms in 2-categories, with special attention to $\mathsf{Cat}$ and $\mathcal{V}$-$\mathsf{Cat}$. We show that any 2-category with convenient colimits has an orthogonal $LaxEpi$-factorization system, and we give a…

Let $\mathsf{T}$ be a triangulated category with shift functor $\Sigma \colon \mathsf{T} \to \mathsf{T}$. Suppose $(\mathsf{A},\mathsf{B})$ is a co-t-structure with coheart $\mathsf{S} = \Sigma \mathsf{A} \cap \mathsf{B}$ and extended…