316 research outputs found
Models of Type Theory Based on Moore Paths
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
Arrow Categories of Monoidal Model Categories
We prove that the arrow category of a monoidal model category, equipped with
the pushout product monoidal structure and the projective model structure, is a
monoidal model category. This answers a question posed by Mark Hovey, and has
the important consequence that it allows for the consideration of a monoidal
product in cubical homotopy theory. As illustrations we include numerous
examples of non-cofibrantly generated monoidal model categories, including
chain complexes, small categories, topological spaces, and pro-categories.Comment: 13 pages. Comments welcome. Version 2 adds more examples, and an
application to cubical homotopy theory. Version 3 is the final, journal
version, accepted to Mathematica Scandinavic
Models of Type Theory Based on Moore Paths
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
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