Related papers: A note on the Penon definition of $n$-category

Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any…

Given an algebraic theory which can be described by a (possibly symmetric) operad $P$, we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for $P$-algebras hold…

We start from any small strict monoidal braided Ab-category and extend it to a monoidal nonstrict braided Ab-category which contains braided bialgebras. The objects of the original category turn out to be modules for these bialgebras

The concept of n-categories and related subject is considered. An n-category is described as an n-graph with a composition. A new definition of operad is presented. Some illustrative examples are given.

We use a 2-categorical version of (de-)equivariantization to classify (3+1)d topological orders with a finite $G$-symmetry. In particular, we argue that (3+1)d fermionic topological order with $G$-symmetry correspond to…

A classification theorem is given of projective threefolds that are covered by a two-dimensional family of lines, but not by a higher dimensional family.

We show that a compact rigid balanced braided monoidal category with enough compact projective objects gives rise to a system of mapping class group representations compatible with the gluing along marked intervals. A motivation to consider…

Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced…

We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples

We show how the categorial approach to inverse monoids can be described as a certain endofunctor (which we call the partialization functor) of some category. In this paper we show that this functor can be used to obtain several recently…

Let $W$ be a finite dimensional purely odd supervector space over $\mathbb{C}$, and let $\sRep(W)$ be the finite symmetric tensor category of finite dimensional superrepresentations of the finite supergroup $W$. We show that the set of…

A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences…

An equivalent description of a symmetric monoidal category is introduced in which, instead of separate associator and commutator isomorphisms satisfying the usual coherence axioms, we simply have associo-commutator isomorphisms satisfying…

An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing them. We survey various concepts of…

We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to…

Due to the work of Shimizu (2019), various nondegeneracy conditions for braided finite tensor categories are equivalent. This theory is partially extended to braided module categories here. We introduce when a braided module category is…

We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and…

We present a general procedure for constructing triangulated categories, linear over a field, with distinct enhancements. Some of our examples can be equipped with a (non-degenerate) t-structure, thereby showing that the existence of a…

We propose a recursive definition of V-n-categories and their morphisms. We show that for V k-fold monoidal the structure of a (k-n)-fold monoidal strict (n+1)-category is possessed by V-n-Cat. This article is a completion of the work begun…

Joyal and Street note in their paper on braided monoidal categories [Braided tensor categories, Advances in Math. 102(1993) 20-78] that the 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in…