The identity type weak factorisation system

TL;DR

This paper demonstrates that the classifying category of a dependent type theory with identity types admits a non-trivial weak factorisation system, linking type theory with homotopy theory of groupoids.

Contribution

It explicitly characterizes the classes in the weak factorisation system and relates identity types to homotopy theory, advancing the understanding of type-theoretic models.

Findings

Classifying category admits a non-trivial weak factorisation system

Explicit characterization of the classes in the system

Connection established between identity types and homotopy theory

Abstract

We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.