Search results for " 03F50"

showing 2 items of 2 documents

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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Lawvere–Tierney sheaves in Algebraic Set Theory

2009

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

Algebraic setPure mathematicsLogicMathematics - Category TheoryMathematics - LogicTopos theoryPhilosophyMathematics::LogicMathematics::Algebraic GeometryMathematics::Category TheoryFOS: MathematicsCategory Theory (math.CT)Algebraic Set Theory sheavesLogic (math.LO)03C90 03G30 03F50AxiomMathematics
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