A functional programmer's guide to homotopy type theory

Dependent type theories are functional programming languages with types rich enough to do computer-checked mathematics and software verification. Homotopy type theory is a recent area of work that connects dependent type theory to the mathematical disciplines of homotopy theory and higher-dimensional category theory. From a programming point of view, these connections have revealed that all types in dependent type theory support a certain generic program that had not previously been exploited. Specifically, each type can be equipped with computationally relevant witnesses of equality of elements of that type, and all types support a generic program that transports elements along these equalities. One mechanism for equipping types with non-trivial witnesses of equality is Voevodsky’s univalence axiom, which implies that equality of types themselves is witnessed by type isomorphism. Another is higher inductive types, an extended datatype schema that allows identifications between different datatype constructors. While these new mechanisms were originally formulated as axiomatic extensions of type theory, recent work has investigated their computational meaning, leading to the development of new programming languages that better support them. In this talk, I will illustrate what univalence and higher inductive types mean in programming terms. I will also discuss how studying some related semantic settings can reveal additional structure on types; for example, moving from groupoids (categories where all maps are invertible) to general categories yields an account of coercions instead of equalities. Overall, I hope to convey some of the beauty and richness of these connections between disciplines, which we are just beginning to understand.