Abstract
AbstractWe 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.
Highlights
Eilenberg and Zilber (1950) used a presheaf model to represent geometrical objects, and the intuition is geometrical: we think of the objects I, J, . . . of the base category as basic “shapes”; a presheaf A is given by a family of sets A(I) of objects of each shape I, which are related by the restriction maps A(I) → A(J)
Scott (1980) described a presheaf model of higher-order logic and pointed out the potential interest for the semantics of λ-calculus. This was refined by Hofmann (1997) who presented a presheaf model of dependent type theory with universes
One contribution of the present paper is to provide a constructive version of this notion1 by describing a sheaf model semantics of type theory with univalence
Summary
The notion of (pre-)sheaf model has a rich and intricate history which mixes different intuitions coming from topology, logic, and algebra. Eilenberg and Zilber (1950) used a presheaf model (simplicial sets) to represent geometrical objects, and the intuition is geometrical: we think of the objects I, J, . . . of the base category as basic “shapes”; a presheaf A is given by a family of sets A(I) of objects of each shape I, which are related by the restriction maps A(I) → A(J). Hofmann’s presheaf model was subsequently used in an essential way in works on constructive semantics of type theory with univalent universes (see Cohen et al 2015; Coquand et al 2018; Orton and Pitts 2016). One contribution of the present paper is to provide a constructive version of this notion by describing a sheaf model semantics of type theory with univalence (see, e.g., Voevodsky 2015 and Univalent Foundations Program 2013). This uses in a crucial way the fact that we have a. We explain how some of our results about descent data operations can be generalized to accessible left-exact modalities
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
