Type Theory and Homotopy

Abstract
of type theory has been used successfully to formalize large parts of constructive mathematics, such as the theory of generalized recursive definitions [NPS90, ML79]. Moreover, it is also employed extensively as a framework for the development of high-level programming languages, in virtue of its combination of expressive strength and desirable proof-theoretic properties [NPS90, Str91]. In addition to simple types A, B, . . . and their terms x : A b(x) : B, the theory also has dependent types x : A B(x), which are regarded as indexed families of types. There are simple type forming operations A × B and A → B, as well as operations on dependent types, including in particular the sum x:A B(x) and product x:A B(x) types (see the appendix for details). The Curry-Howard interpretation of the operations A × B and A → B is as propositional conjunction and..
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Martin-Löf Complexes.S. Awodey & M. A. Warren - 2013 - Annals of Pure and Applied Logic 164 (10):928-956.
Combinatorial Realizability Models of Type Theory.Pieter Hofstra & Michael A. Warren - 2013 - Annals of Pure and Applied Logic 164 (10):957-988.

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