Related papers: A Morphism Double Category and Monoidal Structure
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…
Effectful categories have two classes of morphisms: pure morphisms, which form a monoidal category; and effectful morphisms, which can only be combined monoidally with central morphisms (such as the pure ones), forming a premonoidal…
In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many…
The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $\mathbb A$, the corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category $\mathbb B$…
In many situations one encounters an entity that resembles a monoid. It consists of a carrier and two operations that resemble a unit and a multiplication, subject to three equations that resemble associativity and left and right unital…
We introduce monoidal width as a measure of complexity for morphisms in monoidal categories. Inspired by well-known structural width measures for graphs, like tree width and rank width, monoidal width is based on a notion of syntactic…
We use double categories to obtain a single theorem characterizing certain exponentiable morphisms of small categories, topological spaces, locales, and posets.
We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical…
We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow the category of monoids for the other with…
This is a condensed overview of the formal theory of monads in a 2-category. We also define two double categories of monads in a 2-category, extending Lack and Street's 2-categories of monads.
We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original…
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…
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…
The well-known Lawvere category R of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But R has another such structure, given by multiplication, which is *-autonomous. Normed sets,…
In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally…
We investigate monoidal categories of formal contexts, in which states correspond to formal concepts. In particular we examine the category of bonds or Chu correspondences between contexts, which is known to be equivalent to 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…
We build a model structure from the simple point of departure of a structured interval in a monoidal category - more generally, a structured cylinder and a structured co-cylinder in a category.
We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework is well…
The category of small 2-categories has two monoidal structures due to John Gray: one biclosed and one closed. We propose a formalisation of the construction of the right internal and internal homs of these monoidal structures.