We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a purely type-theoretic contractibility condition which replaces the standard, category-theoretic universal property involving the existence and uniqueness of appropriate morphisms. Our main result characterises the types that are equivalent to W-types as homotopy-initial algebras.
Categories: math.LO math.CT