# Coherent Presentations of Monoidal Categories

## TL;DR

This paper generalizes presentations of categories to include objects modulo an equivalence, establishing conditions under which different constructions of the resulting monoidal categories coincide, extending classical group presentation results.

## Contribution

It introduces a unified framework for presentations of monoidal categories with equational generators, extending known results from group theory to categorical structures.

## Key findings

- The three constructions (quotient, localization, normal forms) coincide under coherence conditions.
- The framework generalizes classical group presentation results to monoidal categories.
- Provides conditions ensuring the equivalence of different categorical constructions.

## Abstract

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations where the objects are considered modulo an equivalence relation, which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one consists in turning equational generators into identities (i.e. considering a quotient category), one consists in formally adding inverses to equational generators (i.e. localizing the category), and one consists in restricting to objects which are normal forms. We show that, under suitable coherence conditions on the presentation, the three constructions coincide, thus generalizing celebrated results on presentations of groups, and we extend those conditions to presentations of monoidal categories.