Related papers: Bicategories in Univalent Foundations
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…
We give a gentle introduction to the concept of folding. That is, we provide an elementary discussion of equivariant categories, their weighted Grothendieck groups, and the technical aspects of computing with them. We then perform the…
We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular…
We define a bicategory in which the 0-cells are the entwinings over variable rings. The 1-cells are triples of a bimodule and two maps of bimodules which satisfy an additional hexagon, two pentagons and two (co)unit triangles; and the…
We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured…
After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky,…
It is well known that to give an oplax functor of bicategories $\mathbf{1}\to\mathscr{C}$ is to give a comonad in $\mathscr{C}$. Here we generalize this fact, replacing the terminal bicategory by any bicategory $\mathscr{A}$ for which the…
We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual…
Holm (H. Holm, Modules with cosupport and injective functors, Algebr. Represent. Theor., 13 (2010), 543-560) considers categories of right modules dual to those with support in a set of finitely presented modules. We extend some of his…
We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution. For this purpose we…
We study monoidal 2-categories and bicategories in terms of categorical extensions and the cohomological data they determine in appropriate cohomology theories with coefficients in Picard groupoids. In particular, we analyze the hierarchy…
We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal…
Covariant Hom-bimodules are introduced and the structure theory of them in the Hom-setting is studied in a detailed way. The category of bicovariant Hom-bimodules is proved to be a (pre)braided monoidal category and its structure theory is…
Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual…
We give the definitions of model bicategory and $q$-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory $\mathcal{C}$, denote by…
In this paper we propose unifying the categories of cochain complexes $\text{Ch}(\mathcal{C})$ and modules $\widehat{A}\text{-mod}$ over a repetitive algebra $\widehat{A}$. Motivated by their striking similarities and importance, we…
A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual "zig-zag" identities of a compact closed category only up to natural isomorphism, and the…
To a B-coring and a (B,A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule…
We give an elementary introduction to the theory of triangulated categories covering their axioms, homological algebra in triangulated categories, triangulated subcategories, and Verdier localization. We try to use a minimal set of axioms…
We introduce a notion of globular multicategory with homomorphism types. These structures arise when organizing collections of "higher category-like" objects such as type theories with identity types. We show how these globular…