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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 $$(\infty , 1)$$(∞,1)-categories with finite limits.

Internal languages of finitely complete $$(\infty , 1)$$ ( ∞ , 1 ) -categories