Constructive sheaf models of type theory

Abstract

We provide a constructive version of the notion of sheaf models of univalent type theory. We start by relativizing existing constructive models of univalent type theory to presheaves over a base category. Any Grothendieck topology of the base category then gives rise to a family of left-exact modalities, and we recover a model of type theory by localizing the presheaf model with respect to this family of left-exact modalities. We provide then some examples.

Unifying cubical and multimodal type theory

“…By assumption, this is a filtered colimit and thus it commutes with finite limits. Since limits are computed pointwise in [C, Set], we may identify (lim X)(c) = lim i X i (c) and compute: We can now prove proposition 4.11: For (1) these topoi satisfy the Orton-Pitts axioms as noted in [CRS21]. To see that the intervals defined above are tiny we proceed as follows: Using the Yoneda lemma along with the fact that I is naturally isomorphic to [−, {i}] shows that y(c, I) × I ∼ = y(c, I + {i}), and we thus calculate:…”

On Church’s thesis in cubical assemblies

“…We will construct a lex modality where Church's thesis is forced to hold, and then use properties of cubical assemblies to show that the modality is non-trivial. Our model can also be viewed as a kind of stack model akin to those used by Coquand, Ruch and Sattler for various independence and consistency results, including the independence of countable choice from homotopy type theory (Coquand et al 2021), although our formulation will be quite different to theirs.…”