A (2,1)-model structure for conceptual completeness

TL;DR

This paper develops a (2,1)-categorical model structure on small coherent categories, extending the small object argument to higher dimensions and establishing a form of conceptual completeness.

Contribution

It introduces a (2,1)-model structure for coherent categories, based on a higher-dimensional reflective factorisation system, and proves properties like right properness.

Findings

Established a (2,1)-model structure on coherent categories

Proved the model structure is right proper

Described the generating trivial cofibrations

Abstract

We prove the (2,1)-categorical analogue of the small object argument and give a (2,1)-model structure on the category of small coherent categories, coherent functors and natural isomorphisms. It is induced by a higher dimensional example of a reflective factorisation system, determined by the full subcategory of pretoposes. We prove it to be right proper and the generating trivial cofibrations are described. Whitehead's theorem gives conceptual completeness.