The (2,1)-category of small coherent categories

TL;DR

This paper explores the (2,1)-category of small coherent categories, establishing its completeness and cocompleteness with respect to 2-limits and 2-colimits, and develops a 2-categorical small object argument for weak factorization systems.

Contribution

It proves that the (2,1)-category of small coherent categories admits all small 2-limits and 2-colimits and introduces a 2-categorical small object argument for coherent functors.

Findings

The category admits all small 2-limits.

The category admits all small 2-colimits.

A 2-categorical small object argument is constructed.

Abstract

There is a well-known correspondence between coherent theories (and their interpretations) and coherent categories (resp. functors), hence the (2,1)-category $\mathbf{Coh_{\sim}}$ (of small coherent categories, coherent functors and all natural isomorphisms) is of logical interest. We prove that this category admits all small 2-limits and 2-colimits (in the ($\infty $,1)-categorical sense), and prove a 2-categorical small object argument to provide weak factorisation systems for coherent functors.