Explicit computational paths in type theory

• Ano de defesa: 2018
• Autor(a) principal: RAMOS, Arthur Freitas
• Orientador(a): OLIVEIRA, Anjolina Grisi de
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de Pernambuco
• Programa de Pós-Graduação: Programa de Pos Graduacao em Ciencia da Computacao
• País: Brasil
• Palavras-chave em Português: Ciência da computação; Teoria da computação
• Link de acesso: https://repositorio.ufpe.br/handle/123456789/32902

The current work has three main objectives. The first one is the proposal of computational paths as a new entity of type theory. In this proposal, we point out the fact that computational paths should be seen as the syntax counterpart of the homotopical paths between terms of a type. We also propose a formalization of the identity type using computational paths. The second objective is the proposal of a mathematical structure fora type using computational paths. We show that using categorical semantics it is possible to induce a groupoid structure for a type and also a higher groupoid structure, using computational paths and a rewrite system. We use this groupoid structure to prove that computational paths also refutes the uniqueness of identity proofs. The last objective is to formulate and prove the main concepts and building blocks of homotopy type theory. We end this last objective with a proof of the isomorphism between the fundamental group of the circle and the group of the integers.