8 research outputs found

    Models of Type Theory Based on Moore Paths

    Full text link
    This paper introduces a new family of models of intensional Martin-L\"of type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.Comment: This is a revised and expanded version of a paper with the same name that appeared in the proceedings of the 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017

    Models of Type Theory Based on Moore Paths

    Get PDF
    This paper introduces a new family of models of intensional Martin-Löf type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and on cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.EPSRC Studentshi

    Internal Universes in Models of Homotopy Type Theory

    Get PDF
    We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory

    Cubical Assemblies, a Univalent and Impredicative Universe and a Failure of Propositional Resizing

    Get PDF

    Relative elegance and cartesian cubes with one connection

    Full text link
    We establish a Quillen equivalence between the Kan-Quillen model structure and a model structure, derived from a model of a cubical type theory, on the category of cartesian cubical sets with one connection. We thereby identify a second model structure which both constructively models homotopy type theory and presents infinity-groupoids, the first known example being the equivariant cartesian model of Awodey-Cavallo-Coquand-Riehl-Sattler.Comment: 60 pages. Comments welcome
    corecore