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This is the text of my talk at CMU on Feb. 4, 2010 were I gave the secondpublic presentation of the Univalence Axiom (called "equivalence axiom" in thetext). The first presentation of the axiom was in a lecture at LMU Munich inNovember 2009.

The equivalence axiom and univalent models of type theory. (Talk at CMU on February 4, 2010)