Abstract

Recent work on higher-dimensional dependent type theory enriches conventional one-dimensional dependent type theory with additional structure expressing equivalence of elements of a type. This structure may be employed in a variety of ways to capture rather coarse identifications of elements that must be respected by type families. Higher-dimensional type theory has applications to code reuse for dependently typed programming, and to the formalization of mathematics. In this paper, we develop a novel judgemental formulation of a two-dimensional type theory, which enjoys a canonicity property: a closed term of boolean type is definitionally equal to one of the two booleans. Canonicity is a necessary condition for a computational interpretation of type theory as a programming language, and does not hold for existing axiomatic presentations of higher-dimensional type theory. The method of proof is a generalization of the NuPRL semantics, interpreting types as syntactic groupoids rather than equivalence relations. 1.

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