Related papers: Dendroidal weak 2-categories
We extend the 2-representation theory of finitary 2-categories to certain 2-categories with infinitely many objects, denoted locally finitary 2-categories, and extend the classical classification results of simple transitive…
In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism…
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond to trihomomorphisms…
We introduce a monoidal category of corings using two different notions of corings morphisms. The first one is the (right) coring extensions recently introduced by T. Brzezi\'nski in [2], and the anther is the usual notion of morphisms…
Weak morphisms of non-abelian complexes of length 2, or crossed modules, are morphisms of the associated 2-group stacks, or gr-stacks. We present a full description of the weak morphisms in terms of diagrams we call butterflies. We give a…
We define and study categories of singular Soergel bimodules, which are certain natural generalisations of Soergel bimodules. Indecomposable singular Soergel bimodules are classified, and we conclude that the split Grothendieck group of the…
We show that doubly degenerate Penon tricategories give symmetric rather than braided monoidal categories. We prove that Penon tricategories cannot give all tricategories, but we show that a slightly modified version of the definition…
Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full…
Cofunctors are a kind of map between categories which lift morphisms along an object assignment. In this paper, we introduce cofunctors between categories enriched in a distributive monoidal category. We define a double category of enriched…
We introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that…
This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck…
We construct explicitly the weights on the simplicial category so that the colimits and limits of 2-functors with those weights provide the Kleisli objects and the Eilenberg-Moore objects, respectively, in any 2-category.
We prove that, under mild assumptions, the $Q$-shaped derived categories introduced by Holm and J{\o}rgensen are equivalent to derived categories of differential graded bimodules over differential graded categories. This yields new derived…
In the literature, one finds several competing notions for the super (i.e., Z/2-graded) analog of a monoidal category. The goal of this paper is to clarify these definitions and the connections between them. We also discuss in detail the…
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…
The notions of weakly bipartitive and bipartitive families were introduced by Montgolfier (2003) as a general tool for studying some decomposition of graphs and other combinatorial structures. In this paper, we give a matrix description of…
We develop the theory of 2-quivers and quiver 2-categories to run in parallel with the classical theory of quiver algebras. A quiver 2-category is always finitary, and, conversely, every finitary 2-category will be bi-equivalent with a…
An extension of order theory is presented that serves as a formalism for the study of dendroidal sets analogously to way the formalism of order theory is used in the study of simplicial sets.
We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two. These braidings are shown to arise from, and classify, cobraidings (also known…
We develop the Witt group for certain braided monoidal categories with duality. In case of a braided fusion category over an algebraically closed field of characteristic zero, we explicitly describe this structure. We then use this…