The HoTT Library: A formalization of homotopy type theory in Coq

Data structures for quasistrict higher categories

“…This broad research programme seeks to develop a new foundation for mathematics, in a setting based on a homotopical interpretation of Martin-Löf's dependent type theory [51]. It is amenable to computer formalization, and there has been substantial activity in producing formal proofs [1,11,37]. While the motivations are related, there is little overlap between proofs suitable for formalization in homotopy type theory, and those suitable for formalization in our system.…”

“…Extended variant. The type II homotopy generators include 2 further variants of the moves shown in (11), corresponding to pulling a vertex through an inverse type I homotopy generator, as well as the corresponding composition schemes.…”

An introduction to univalent foundations for mathematicians

“…Luckily, it is a theorem of Voevodsky that, for any type X, the type isSet(X) is a proposition. 17 Applied to G, it shows that ι = ι ′ , implying in turn that our two groups are equal. This fortuitous foundational result helps to show the feasibility of the approach.…”

“…With equality types available, with all their expected properties, we may encode some elementary mathematical properties as types, to show how such encoding goes in practice, as implemented (approximately) in: the UniMath project [72], which is exposed by Voevodsky in [64]; as in the HoTT project [4], which is exposed in [17]; as in the HoTT-Agda project [1]; and as in the Lean theorem prover [2].…”

“…Now let's consider the problem of formalizing the notion of G-torsor. By definition, it is a nonempty set X with a free transitive 18 left action 17 His proof depends on two axioms, both of which are consequences of the Univalence Axiom to be presented later. One of them is a strong form of function extensionality, which states that functions having equal values are equal.…”