Related papers: Strictification of categories weakly enriched in s…

We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that…

This paper provides a comprehensive overview of some of the foundational properties of categories enriched over quantaloids, along with several new results. We demonstrate that the category whose objects are quantaloid-enriched categories…

In this survey paper we give account of several approaches to the strictification and non-strictification of monoidal categories, which are constructions that turn a monoidal category into a (non-)strict one monoidally equivalent to the…

We show that some recent constructions in the literature, named `weak' generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.

We show that for a monoidal model category $\M=(\ul{M}, \otimes, I)$, certain co-Segal $\M$-categories are equivalent to strict ones.

It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a…

We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to the cartesian monoidal structure. As…

Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between…

This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the…

This paper considers the possible underlying multicategories for a symmetric monoidal category, and shows that, up to canonical and coherent isomorphism, there really is only one. As a result, there is a well-defined forgetful functor from…

This thesis is devoted to the proof of a theorem showing the existence of a closed model category structure for weakly enriched categories. It requires first of all the definitions of weakly enriched categories and equivalences of weakly…

We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory.

A multicategory is what remains of a monoidal category when monoidal product is not available. A weak multicategory means that hom-sets are in fact categories, and in place of usual equations, there are natural isomorphisms, which have to…

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 continue our study of semi-strict tricategories in which the only weakness is in vertical composition. We assemble the doubly-degenerate such tricategories into a 2-category, defining weak functors and transformations. We exhibit a…

We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad -- a structure with 2-cells of all possible double-categorical shapes --…

Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided…

Lyubashenko has described enriched 2-categories as categories enriched over V-Cat, the 2-category of categories enriched over a symmetric monoidal V. I have generalized this to the k-fold monoidal V. The symmetric case can easily be…

Braided-enriched monoidal categories were introduced in work of Morrison-Penneys, where they were characterized using braided central functors. Recent work of Kong-Yuan-Zhang-Zheng and Dell extended this characterization to an equivalence…

Poly-bicategories generalise planar polycategories in the same way as bicategories generalise monoidal categories. In a poly-bicategory, the existence of enough 2-cells satisfying certain universal properties (representability) induces…