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

I will show how to define, in any type system with dependent sums, products and Martin-Lof identity types, the notion of a homotopy equivalence between two types and how to formulate the Equivalence Axiom which provides a natural way to assert that ”two homotopy equivalent types are equal”. I will then sketch a construction of a model of one of the standard Martin-Lof type theories which satisfies the equivalence axiom and the excluded middle thus proving that M.L. type theory with excluded middle and equivalence axiom is at least as consistent as ZFC theory. Models which satisfy the equivalence axiom are called univalent. This is a totally new class of models and I will argue that the semantics which they provide leads to the first satisfactory approach to typetheoretic formalization of mathematics. 1 Formal deduction systems and quasi-equational theories. Type systems and conservative extensions of the theory of contextual categories. I will speak about type systems. It is difficult for a mathematician since a type system is not a mathematical notion. I will spend a little time explaining how I see ”type systems” mathematically. ”Type systems” are formal deduction systems of particular ”flavor”. So let me start with the following: Thesis 0. Any formal deduction system can be specified in the form of a quasi-equational theory. Quasi-equational theories are multi-sorted algebraic theories whose operations are given together with an ordering and ”domains of definitions” which are specified by equations involving preceding operations. It is a very important class of theories. In particular, algebraic theories (e.g. groups with given relations) are quasi-equational. The classic example of a properly quasi-equational theory is the theory of set-level categories i.e. categories