On the identity type as the type of computational paths

Abstract

We introduce a new way of formalizing the intensional identity type based on the fact that a entity known as computational paths can be interpreted as terms of the identity type. Our approach enjoys the fact that our elimination rule is easy to understand and use. We make this point clear constructing terms of some relevant types using our proposed elimination rule. We also show that the identity type, as defined by our approach, induces a groupoid structure. This result is on par with the fact that the traditional identity type induces a groupoid, as exposed by Hofmann & Streicher (1994).

Conversão de Termos, Homotopia, e Estrutura de Grupóide

“…Como já mencionado anteriormente, os caminhos da interpretação homotópica das diferentes formulações da teoria de tipos de Martin-Löf não têm contrapartida na sintaxe, mas existem apenas na semântica. Adicionamosà sintaxe termos especificamente definidos para representar os caminhos computacionais e ainda assim permanecemos em consonância com os princípios da interpretação de Curry-Howard, pois esses termos representam provas de igualdade [de Queiroz et al 2016, Ramos et al 2017, de Queiroz and de Oliveira 2014.…”

“…Dessa forma, podemos considerar uma estrutura que para cada par de pontos (a, b) ∈ Π(A), existe uma categoria fraca formada pelos caminhos entre os caminhos que conectam a e b. Essa estrutura seria o grupóide fundamental de segunda ordem do tipo A, Π 2 (A). Provar que essa estruturaé realmente um grupóide de segunda ordem fica fora do escopo deste trabalho, mas a prova completa pode ser vista em [Ramos et al 2017]. A linha de raciocínio envolvida nesse tipo de trabalhoé ilustrada por [Awodey 2012, Awodey 2017] na exposição das conexões entre teoria dos tipos e teoria da homotopia (see Fig.…”

Explicit Computational Paths in Type Theory

“…Thus, we conclude our review of computational paths as terms of the identity type and the associated rewrite system. If necessary, please check [7] and [14] for a thorough development of this theory.…”

“…In fact, we proposed in [7] that it should be considered as the type of the identity type. Moreover, we have further developed this idea in [14], where we proposed a groupoid model and proved that computational paths also refute the uniqueness of identity proofs. Thus, we obtained a result that is on par with the same one obtained by Hofmann & Streicher (1995) for the original identity type [12].…”

“…In section 5, we use computational paths to establish the foundations of homotopy type theory. Since sections 2, 3 and 4 are only brief introductions to the theory of computational paths, we refer to papers [7] and [14] for a thoroughly introduction to this subject.…”