Search results for "Homotopy type theory"

showing 2 items of 2 documents

The identity type weak factorisation system

2008

We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.

Class (set theory)Pure mathematicsGeneral Computer ScienceDependent type theoryHomotopiaType (model theory)Identity (music)Theoretical Computer Science510 - Consideracions fonamentals i generals de les matemàtiquesCombinatorics18C50Mathematics::Category TheoryFOS: MathematicsCategory Theory (math.CT)Univalent foundationsAxiomMathematicsHomotopy03B15; 18C50; 18B40Mathematics - Category TheoryIdentity type weak factorisation systemMathematics - LogicTipus Teoria dels03B15Type theory18B40Homotopy type theoryLogic (math.LO)Weak factorisation systemIdentity typeComputer Science(all)
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Inductive types in homotopy type theory

2012

Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for intensional systems of type theory as well as a computational approach to algebraic topology via type theory-based proof assistants such as Coq. The present work investigates inductive types in this setting. Modified rules for inductive types, including types of well-founded trees, or W-types, are presented, and the basic homotopical semantics of such types are determined. Proofs of all results have been formally verified by the Coq proof assistant, and the proof s…

FOS: Computer and information sciencesComputer Science - Logic in Computer Science03B15 03B70 03F500102 computer and information sciences01 natural sciencesComputer Science::Logic in Computer ScienceFOS: MathematicsA¹ homotopy theoryCategory Theory (math.CT)0101 mathematicsMathematicsHomotopy lifting propertyType theory inductive types homotopy-initial algebraHomotopy010102 general mathematicsMathematics - Category TheoryIntuitionistic type theoryMathematics - LogicSettore MAT/01 - Logica MatematicaLogic in Computer Science (cs.LO)Algebran-connectedType theoryTheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES010201 computation theory & mathematicsProof theoryTheoryofComputation_LOGICSANDMEANINGSOFPROGRAMSHomotopy type theoryComputer Science::Programming LanguagesLogic (math.LO)
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