Internal Languages of Finitely Complete $(\infty ,1)$-categories

Abstract

We prove that the homotopy theory of Joyal's tribes is equivalent to that of fibration categories. As a consequence, we deduce a variant of the conjecture asserting that Martin-Löf Type Theory with dependent sums and intensional identity types is the internal language of $(\infty ,1)$-categories with finite limits.

41 pages, minor revisions