Identity in Homotopy Type Theory, Part I: The Justification of Path Induction

Homotopy type theory (HoTT) is a new branch of mathematics that connects algebraic topology with logic and computer science, and which has been proposed as a new language and conceptual framework for mathematical practice. Much of the power of HoTT lies in the correspondence between the formal type theory and ideas from homotopy theory, in particular the interpretation of types, tokens, and equalities as (respectively) spaces, points, and paths. Fundamental to the use of identity and equality in HoTT is the powerful proof technique of path induction. In the ‘HoTT Book’ [1] this principle is justified through the homotopy interpretation of type theory, by treating identifications as paths and the induction step as a homotopy between paths. This is incompatible with HoTT being an autonomous foundation for mathematics, since any such foundation must be able to justify its principles without recourse to existing areas of mathematics. In this paper it is shown that path induction can be motivated from pre-mathematical considerations, and in particular without recourse to homotopy theory. This makes HoTT a candidate for being an autonomous foundation for mathematics.