typetheory
Type theory is a family of formal systems in which every expression has a type, and computation proceeds by manipulating typed terms. In type theories, types classify terms and prevent ill-formed expressions; functions are defined by specifying how inputs of a given type produce outputs of another type. A distinctive feature of modern type theory is the possibility that types themselves may depend on terms, enabling precise, expressive encodings of mathematical objects and propositions.
A central idea is the Curry–Howard correspondence, which identifies propositions with types and proofs with programs.
Core systems include the simply typed lambda calculus, which enforces a fixed type structure, and its extensions
In recent decades, Homotopy Type Theory (HoTT) and Univalent Foundations have integrated ideas from algebraic topology