Related papers: Constructing symmetric monoidal bicategories funct…
Actions of monoidal categories on categories, also known as actegories, have been familiar to category theorists for a long time, and yet a comprehensive overview of this topic seems to be missing from the literature. Recently, actegories…
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 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.
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…
Milner's bigraphs are a general framework for reasoning about distributed and concurrent programming languages. Notably, it has been designed to encompass both the pi-calculus and the Ambient calculus. This paper is only concerned with…
A symmetric monoidal category is a category equipped with an associative and commutative (binary) product and an object which is the unit for the product. In fact, those properties only hold up to natural isomorphisms which satisfy some…
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external…
We study a functorial construction from the category of monoids to the category of set-operads and we give some combinatorial examples of applications.
We define the phrase `category enriched in an fc-multicategory' and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal…
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…
This article is devoted to the investigation of the deformation (twisting) of monoidal structures, such as the associativity constraint of the monoidal category and the monoidal structure of monoidal functor. The sets of twistings have a…
Baez-Dolan type plus constructions serve three main purposes: They (1) corepresent categorical bimodules that are monoids with respect to a plethysm product, (2) allow to define functors as bimodule monoids, and thereby algebras over…
Motivated by viewing categories as bimodule monoids over their isomorphism groupoids, we construct monoidal structures called plethysm products on three levels: that is for bimodules, relative bimodules and factorizable bimodules. For the…
Given a category with a bifunctor and natural isomorphisms for associativity, commutativity and left and right identity we do not assume that extra constraining diagrams hold. We introduce groupoids of coupling trees to describe a version…
Given a monoidal $\infty$-category $C$ equipped with a monoidal recollement, we give a simple criterion for an object in $C$ to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them.…
We show that the category of numerically generated pointed spaces is complete, cocomplete, and monoidally closed with respect to the smash product, and then utilize these features to establish a simple but flexible method for constructing…
We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this…
A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity…
One aim of this paper is to develop some aspects of the theory of monoidal derivators. The passages from categories and model categories to derivators both respect monoidal objects and hence give rise to natural examples. We also introduce…
In this note, we define an analogue of R-matrices for bialgebras in the setting of a monad that is opmonoidal over two tensor products. Analogous to the classical case, such structures bijectively correspond to duoidal structures on the…