Related papers: Cartesian Gray-Monoidal Double Categories
When designing plans in engineering, it is often necessary to consider attributes associated to objects, e.g. the location of a robot. Our aim in this paper is to incorporate attributes into existing categorical formalisms for planning,…
Algebraic structures such as monoids, groups, and categories can be formulated within a category using commutative diagrams. In many common categories these reduce to familiar cases. In particular, group objects in Grp are abelian groups,…
We show how the notion of intercategory encompasses a wide variety of three-dimensional structures from the literature, notably duoidal categories, monoidal double categories, cubical bicategories, double bicategories and Gray categories.…
We present a method of constructing symmetric monoidal bicategories from symmetric monoidal double categories that satisfy a lifting condition. Such symmetric monoidal double categories frequently occur in nature, so the method is widely…
We show that the category of (reflexive) graphs and graph maps carries exactly two closed symmetric monoidal products: the box product and the categorical product.
We show that braided, sylleptic and symmetric monoidal bicategories are precisely the $\mathsf{E}_k$-monoids in the cartesian monoidal $(\infty,1)$-category of bicategories for respective integers $k$. To manage the underlying computations,…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
In this expository paper, we discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
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…
This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a…
It is well-known that small categories have equivalent descriptions as partial monoids. We provide a formulation of partial monoid and partial monoid homomorphism involving $s$ and $t$ instead of identities and then following a recent…
We construct so called Hall monoidal categories (and Hall modules thereover) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are…
Let $\mathcal C$ be a category with finite colimits, writing its coproduct $+$, and let $(\mathcal D, \otimes)$ be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided monoidal…
The category of Cartesian cubical sets is introduced and endowed with a Quillen model structure using ideas coming from recent constructions of cubical systems of univalent type theory.
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…
Applied category theory often studies symmetric monoidal categories (SMCs) whose morphisms represent open systems. These structures naturally accommodate complex wiring patterns, leveraging (co)monoidal structures for splitting and merging…
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a…
Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the…
For each $n \in \mathbb{N} \cup \{\infty\}$, diagrammatic sets admit a model structure whose fibrant objects are the diagrammatic $(\infty, n)$- categories. They also support a notion of Gray product given by the Day convolution of a…