Two-dimensional models of type theory

This is the second in a series of papers detailing the author’s investigations into the intensional type theory of Martin-Löf, as described in Nordström et al. (1990). The first of these papers, Garner (2009), investigated syntactic issues relating to its dependent product types. The present paper is a contribution to its categorical semantics. Seely (1984) proposed that the correct categorical models for extensional Martin-Löf type theory should be locally cartesian closed categories: these being categories C with finite limits in which each of the functors f∗ : C/X → C/Y induced by pulling back along a morphism f : Y → X has a right adjoint. The idea is to think of each object X of a locally cartesian closed category C as a closed type, each morphism as a term and each object of the slice category C/X as a type dependent upon X. Now substitution of terms in types may be interpreted by pullback between the slices of C, dependent sum and product types by left and right adjoints to pullback and the equality type on X by the diagonal morphism Δ: X → X × X in C/X × X. It was later pointed out in Hofmann (1995b) that this picture, whilst very appealing, is not wholly accurate, since in the syntax, the operation that assigns to each morphism of types f : Y → X the corresponding substitution operation Type(X) → Type(Y ) is strictly functorial in f; whilst in the semantics, the corresponding assignation (f : Y → X) → (f∗ : C/X → C/Y ) is rarely so. Thus, this notion of model is not sound for the syntax, and we are forced to refine it slightly: essentially by equipping our locally cartesian closed category with a split fibration T → C equivalent to its codomain fibration C → C. Types over X are now interpreted as objects of the fibre category T(X); and since T → C is a split fibration, the interpretation is sound for substitution. The question of how the above should generalise from extensional to intensional Martin-Löf type theory is a delicate one. It is possible to paraphrase the syntax of intensional type theory in categorical language and so arrive at a notion of model – as