Category Theory
Homotopy Type Theory
Vladimir Voevodsky in 2009 formulated the univalence axiom and proved its consistency via simplicial semantics. In 2013 the Institute for Advanced Study published 'Homotopy Type Theory', written by a collaboration of 50 mathematicians in one year.
- Formal mathematics: Lean 4, Agda, Coq -- proof assistants based on dependent type theory
- Cubical computational semantics: Cubical Agda checks HoTT theorems computationally
- Infinity-categories: HoTT is a synthetic theory of infinity-groupoids, analogous to Lurie's theory without model categories
Предварительные знания
Types as Infinity-Groupoids
Homotopy Type Theory began with Voevodsky's 2006 univalence axiom and crystallized in the 2013 HoTT Book written collaboratively at the IAS Univalent Foundations Program. Coq and Lean formalizations of HoTT now check proofs spanning thousands of pages, and the cubical type theory implementation cubicaltt (2016) computes the univalence axiom directly.
In HoTT every type A has an h-level. A is contractible (h-level -2) if there exists a_0 such that for all a: a = a_0. A is a proposition (h-level -1) if for all a b: A, a = b. A is a set (h-level 0) if all paths a = b are contractible. H-level n+1: all Id_A(a,b) have h-level at most n. Univalence axiom: (A = B) ~= (A ~= B).
What is the h-level of S^1?
Higher Inductive Types
A higher inductive type (HIT) is specified by point and path constructors. S^1 = {base : S^1; loop : base = base}. The integers Z are built as the cover of S^1. n-truncation ||A||_n adds a constructor forcing h-level at most n. Suspension SA, join A * B are HITs. Cubical type theory makes univalence and HITs computationally well-defined.
What specifies S^1 in HoTT as a HIT?
Key Ideas
- Types are infinity-groupoids: h-level(A) measures path space complexity
- Univalence: (A = B) ~= (A ~= B) -- isomorphic types are identical
- pi_1(S^1) = Z: proved in HoTT via encode-decode without topology
- HITs: S^1 = {base, loop}, suspension, truncation -- algebraic constructions
- Cubical TT: univalence is computational, HITs have definitional semantics
Further Directions
These ideas open paths to deeper mathematics.
- ct-28-model-categories — extends
Вопросы для размышления
- Give a concrete example.
- How does this connect to other areas of mathematics?