Higher Inductive Types in Homotopy Type Theory

Homotopy Type Theory (HoTT) refers to the homotopical interpretation [1] of Martin-Löf’s intensional, constructive type theory (MLTT) [5], together with several new principles motivated by that interpretation. Voevodsky’s Univalent Foundations program [6] is a conception for a new foundation for mathematics, based on HoTT and implemented in a proof assistant like Coq [2]. Among the new principles to be added to MLTT are the Univalence Axiom [4], and the so-called higher inductive types (HITs), a new idea due to Lumsdaine and Shulman which allows for the introduction of some basic spaces and constructions from homotopy theory. For example, the n-dimensional spheres S can be implemented as HITs, in a way analogous to the implementation of the natural numbers as a conventional inductive type. Other examples include the unit interval; truncations, such as bracket-types [A]; and quotients by equivalent relations or groupoids. The combination of univalence and HITs is turning out to be a very powerful and workable system for the formalization of homotopy theory, with the recently given, formally verified proofs of some fundamental results, such as determinations of various of the homotopy groups of spheres by Brunerie and Licata. See [3] for much work in progress After briefly reviewing the foregoing developments, I will give an impredicative encoding of certain HITs on the basis of a new representation theorem, which states that every type of a particular kind is equivalent to its double dual in the space of coherent natural transformations. A realizability model is also provided, establishing the consistency of impredicative HoTT and its extension by HITs.