A survey of constructive presheaf models of univalence

Abstract

Any formal system for representing mathematics should address the two questions of how to represent collections of mathematical objects and how to decide the laws of identifications of these objects. These laws of identifications have become quite subtle. While it has been clear for a long time that it is good mathematical practice to identify isomorphic algebraic structures [11], or at least to use only notions and facts about algebraic structures that are invariant under isomorphisms, category theory extends this to the notion of categorical equivalences 1 , which themselves have been generalized to higher forms of equivalences [25]. Voevodsky noticed that, by extending some versions of dependent type theory with one further axiom - the univalence axiom - one obtains a formal system in which all notions and operations are automatically invariant under isomorphisms and even under higher notions of equivalence.