Related papers: Weak vertical composition II: totalities
This paper introduces the construction of a weakly globular double category of fractions for a category and studies its universal properties. It shows that this double category is locally small and considers a couple of concrete examples.
For a weak 2-group, we construct a bicategory of flat 2-group bundles over differentiable stacks as a localization of a functor bicategory. This description is amenable to explicit geometric constructions. For example, we show that flat…
We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors…
After two papers on weak cubical categories and {\it collarable} cospans, respectively, we put things together and construct a {\it weak} cubical category of cubical {\it collared} cospans of topological spaces. We also build a second…
We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories. This…
Let G be a finite group. Given a finite G-set X and a modular tensor category C, we construct a weak G-equivariant fusion category, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of…
We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to…
We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully…
In this paper we redevelop the foundations of the category theory of quasi-categories (also called infinity-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among…
Whereas formal category theory is classically considered within a $2$-category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of augmented virtual double categories, a notion…
This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. The two main…
The notion of a weak duality involution on a bicategory was recently introduced by Shulman in [arXiv:1606.05058]. We construct a weak duality involution on the fully dualisable part of $\text{Alg}$, the Morita bicategory of…
We reprove the classical Tannaka-Krein reconstruction theorem by finding a monoidal equivalence of categories between a 1-truncated sub-2-category of the slice 2-category ${\sf Mod}({\sf Vec})/{\sf Vec}$ and the category of algebras. We…
We develop a general theory of (extended) inner autoequivalences of objects of any 2-category, generalizing the theory of isotropy groups to the 2-categorical setting. We show how dense subcategories let one compute isotropy in the presence…
In this article, we investigate monoidal, braided, sylleptic centralizers of monoidal, braided, sylleptic 2-functors. We specifically focus on multifusion 2-categories and show that monoidal, braided, sylleptic centralizers are multifusion…
Classical definitions of weak higher-dimensional categories are given inductively; for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent…
The theory of 2-monads entails that, for a strict monoidal category C, there is a strict monoidal category L(C) such that strict monoidal functors from L(C) are precisely the lax monoidal functors from C. We give an elementary,…
We develop a theory of weak omega categories that will be accessible to anyone who is familiar with the language of categories and functors and who has encountered the definition of a strict 2-category. The most remarkable feature of this…
In this note we prove that additive categories that occur as hearts of weight structures are precisely the weakly idempotent completecategories, that is, the categories where all split monomorphisms give direct sum decompositions. We also…
There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened -- these are tentatively called fair…