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Abstract: 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 data
