Homotopy Type Theory

Homotopy type theory unifies mathematical proof and computation: a proof is a program, a type is a specification. Voevodsky's univalence axiom makes isomorphic structures identical - eliminating the distinction between 'equality' and 'isomorphism' in mathematics.

• Lean 4 / Mathlib4: 250,000 formally verified lemmas including the Fermat-Wiles theorem - a direct application of type theory based on HoTT principles
• Agda with cubical type theory (Agda Cubical): the first proof assistant with computational univalence, where pi_1(S^1) = Z is verified as a running program
• Synthetic algebraic geometry: formalization of Grothendieck schemes in HoTT is an active area in the UniMath library for Coq
• Dependent types in programming languages (Idris, Agda): direct application of Martin-Lof type theory for verifying programs with precise invariants

HoTT: Types, Paths, and Univalence

Vladimir Voevodsky in 2009 proposed the univalence axiom, combining Martin-Lof type theory with homotopy topology. The HoTT book (2013) was written jointly by 50 mathematicians in one year. Lean 4 (Mathlib4, 250,000 lemmas) and Coq use type theories based on the same principles.

Higher Inductive Types and Truncations

Higher inductive types (HIT) allow defining spaces axiomatically: specifying points and paths between them as constructors. Truncations allow 'forgetting' path structure above a given level - turning an arbitrary type into a proposition, a set, or a groupoid.

HoTT opens up synthetic homotopy theory: classical theorems of algebraic topology (fundamental group of S^1, Hopf fibration theorem) are formally proved in Agda and Lean proof assistants without classical topology. Proof and theorem become a single computational object.

Итоги

• Identity type a =_A b: type of equality proofs, elements are paths; structure of an infinity-groupoid
• Univalence axiom: (A ~ B) ~ (A = B) - equivalence of types is equivalent to their equality in the universe
• Higher inductive types (HIT): path constructors allow axiomatic definitions of spaces (S^1, T^2, K(G,1))
• Transport: a path p: a = b automatically carries proofs of predicate P(a) to proofs of P(b)
• Propositional truncation ||A||_{-1}: turns a type into the proposition 'A is inhabited' without a specific element
• Cubical TT (Agda Cubical): univalence becomes computational via interval type I = {0, 1, seg}