Abstract
We construct a left semi-model structure on the category of intensional type theories (precisely, on CxlCatId,1,Σ(,Πext)). This presents an ∞-category of such type theories; we show moreover that there is an ∞-functor Cl∞ from there to the ∞-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.
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