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

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 $C/M$.