The HoTT library: a formalization of homotopy type theory in Coq

Abstract

We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant. It formalizes most of basic homotopy type theory, including univalence, higher inductive types, and signi cant amounts of synthetic homotopy theory, as well as category theory and modalities. The library has been used as a basis for several independent developments. We discuss the decisions that led to the design of the library, and we comment on the interaction of homotopy type theory with recently introduced features of Coq, such as universe polymorphism and private inductive types.

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.…”

The RedPRL Proof Assistant (Invited Paper)

“…Homotopy type theory [27] and Univalent Foundations [29] extend traditional type theory with a number of axioms inspired by homotopy-theoretic models [19], namely Voevodsky's univalence axiom [28] and higher inductive types [21]. In recent years, these systems have been deployed as algebraic frameworks for formalizing results in synthetic homotopy theory [8,12,17], sometimes even leading to the discovery of novel generalizations of classical theorems [1].…”

Homotopy Type Theory in Lean

“…Most previous formalizations of HoTT used proof assistants that were not originally designed with the homotopy interpretation in mind. In Coq we have both Voevodsky et al's UniMath project [23] and the HoTT library [4]. In Agda, there is another substantial HoTT library [5].…”