15-819 Homotopy Type Theory Lecture Notes

Recall from previous lectures the definitions of functionality and transport. Functionality states that functions preserve identity; that is, domain elements equal in their type map to equal elements in the codomain. Transportation states the same for type families. Traditionally, this means that if a =A a′, then B[a] true iff B[a′] true. In proof-relevant mathematics, this logical equivalence is generalized to a statement about identity in the family: if a =A a′, then B[a] =B B[a′]. Transportation can be thought of in terms of functional extensionality. Unfortunately, extensionality fails in ITT. One way to recover extensionality, which comports with traditional mathematics, is to reduce all identity to reflexivity. This approach, called Extensional Type theory (ETT), provides a natural setting for set-level mathematics. The HoTT perspective on ETT is that the path structure of types need not be limited to that of strict sets. The richer path structure of an ∞-groupoid is induced by the induction principle for identity types. Finding a type-theoretic description of this behavior (that is, introduction, elimination and computation rules which comport with Gentzen’s Inversion Principle) is an open problem.