In mathematical logic and computer science, homotopy type theory (HoTT) includes various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homotopical and higher-categorical models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants. There is a large overlap between the work referred to as homotopy type theory, and that called the univalent foundations project. Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also sometimes corresponds to differences in viewpoint and emphasis. As such, this article may not represent the views of all researchers in the fields equally. This kind of variability is unavoidable when a field is in rapid flux.
History
Groupoid model At one time, the idea that types in intensional type theory with their identity types could be regarded as groupoids was mathematical folklore. It was first made precise semantically in the 1994 paper of Martin Hofmann and Thomas Streicher called "The groupoid model refutes uniqueness of identity proofs", in which they showed that intensional type theory had a model in the category of groupoids. This was the first truly "homotopical" model of type theory, albeit only "1-dimensional" (the traditional models in the category of sets being homotopically 0-dimensional). Their follow-up paper foreshadowed several later developments in homotopy type theory. For instance, they noted that the groupoid model satisfies a rule they called "universe extensionality", which is none other than the restriction to 1-types of the univalence axiom that Vladimir Voevodsky proposed 10 years later. (The axiom for 1-types is notably simpler to formulate, however, since a coherent notion of "equivalence" is not required.) They also defined "categories with isomorphism as equality" and conjectured that in a model using higher-dimensional groupoids, for such categories one would have "equivalence is equality"; this was later proven by Benedikt Ahrens, Krzysztof Kapulkin, and Michael Shulman.
Early history: model categories and higher groupoids The first higher-dimensional models of intensional type theory were constructed by Steve Awodey and his student Michael Warren in 2005 using Quillen model categories. These results were first presented in public at the conference FMCS 2006 at which Warren gave a talk titled "Homotopy models of intensional type theory", which also served as his thesis prospectus (the dissertation committee present were Awodey, Nicola Gambino and Alex Simpson). A summary is contained in Warren's thesis prospectus abstract. At a subsequent workshop about identity types at Uppsala University in 2006 there were two talks about the relation between intensional type theory and factorization systems: one by Richard Garner, "Factorisation systems for type theory", and one by Michael Warren, "Model categories and intensional identity types". Related ideas were discussed in the talks by Steve Awodey, "Type theory of higher-dimensional categories", and Thomas Streicher, "Identity types vs. weak omega-groupoids: some ideas, some problems". At the same conference Benno van den Berg gave a talk titled "Types as weak omega-categories" where he outlined the ideas that later became the subject of a joint paper with Richard Garner. All early constructions of higher dimensional models had to deal with the problem of coherence typical of models of dependent type theory, and various solutions were developed. One such was given in 2009 by Voevodsky, another in 2010 by van den Berg and Garner. A general solution, building on Voevodsky's construction, was eventually given by Lumsdaine and Warren in 2014. At the PSSL86 in 2007 Awodey gave a talk titled "Homotopy type theory" (this was the first public usage of that term, which was coined by Awodey). Awodey and Warren summarized their results in the paper "Homotopy theoretic models of identity types", which was posted on the ArXiv preprint server in 2007 and published in 2009; a more detailed version appeared in Warren's thesis "Homotopy theoretic aspects of constructive type theory" in 2008. At about the same time, Vladimir Voevodsky was independently investigating type theory in the context of the search of a language for practical formalization of mathematics. In September 2006 he posted to the Types mailing list "A very short note on homotopy lambda calculus", which sketched the outlines of a type theory with dependent products, sums and universes and of a model of this type theory in Kan simplicial sets. It began by saying "The homotopy λ-calculus is a hypothetical (at the moment) type system" and ended with "At the moment much of what I said above is at the level of conjectures. Even the definition of the model of TS in the homotopy category is non-trivial" referring to the complex coherence issues that were not resolved until 2009. This note included a syntactic definition of "equality types" that were claimed to be interpreted in the model by path-spaces, but did not consider Per Martin-Löf's rules for identity types. It also stratified the universes by homotopy dimension in addition to size, an idea that later was mostly discarded. On the syntactic side, Benno van den Berg conjectured in 2006 that the tower of identity types of a type in intensional type theory should have the structure of an ω-category, and indeed a ω-groupoid, in the "globular, algebraic" sense of Michael Batanin. This was later proven independently by van den Berg and Garner in the paper "Types are weak omega-groupoids" (published 2008), and by Peter Lumsdaine in the paper "Weak ω-Categories from Intensional Type Theory" (published 2009) and as part of his 2010 Ph.D. thesis "Higher Categories from Type Theories".
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