Combinatorial realizability models of type theory

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Abstract

We introduce a new model construction for Martin-Löf intensional type theory, which is sound and complete for the 1-truncated version of the theory. The model formally combines, by gluing along the functor from the category of contexts to the category of groupoids, the syntactic model with a notion of realizability. As our main application, we use the model to analyse the syntactic groupoid associated to the type theory generated by a graph G, showing that it has the same homotopy type as the free groupoid generated by G.

MSC

03B15
03G30
18A15
18D30

Keywords

Realizability
Gluing
Logical relations
Homotopy type theory
Martin-Löf type theory
Groupoid semantics

Cited by (0)

1

During the preparation of this work Hofstra received support from an NSERC Discovery Grant.

2

During the preparation of this work Warren received support from the Fields Institute and NSF Grant DMS-0635607.