The Quotient of a Category by the Action of a Monoidal Category

TL;DR

This paper defines and constructs the quotient of a category by a monoidal category action, resulting in a 2-category, and proves its existence and uniqueness through generators and relations.

Contribution

It introduces the concept of the quotient of a category by a monoidal action and provides a method to present it via generators and relations.

Findings

The quotient $C/M$ is a well-defined 2-category.

Existence and uniqueness of the quotient are established.

A presentation of the quotient using generators and relations is provided.

Abstract

We introduce the notion of the quotient of a category $C$ by the action $A : M \longrightarrow C \times C$ of a unital symmetric monoidal category $M$. The quotient $C/M$ is a 2-category. We prove its existence and uniqueness by first showing that every small 2-category has a presentation in terms of generators and relations and then describing the generators and relations needed for the quotient $C/M$.