The homotopy theory of type theories

Abstract

We construct a left semi-model structure on the category of intensional type theories (precisely, on $\mathrm{CxlCat_{Id,1,\Sigma(,\Pi_{ext})}}$). This presents an $\infty$-category of such type theories; we show moreover that there is an $\infty$-functor $\mathrm{Cl}_\infty$ from there to the $\infty$-category of suitably structured quasi-categories. This allows a precise formulation of the conjectures that intensional type theory gives internal languages for higher categories, and provides a framework and toolbox for further progress on these conjectures.