Constructive sheaf models of type theory

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.

Coarse cohomology with twisted coefficients

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

Quotients of a theory of presheaf type

In this chapter the quotients of a given theory of presheaf type are investigated by means of Grothendieck topologies that can be naturally attached to them, establishing a ‘semantic’ representation for the classifying topos of such a quotient as a subtopos of the classifying topos of the given theory of presheaf type. It is also shown that the models of such a quotient can be characterized among the models of the theory of presheaf type as those which satisfy a key property of homogeneity with respect to a Grothendieck topology associated with the quotient. A number of sufficient conditions for the quotient of a theory of presheaf type to be again of presheaf type are also identified: these include a finality property of the category of models of the quotient with respect to the category of models of the theory and a rigidity property of the Grothendieck topology associated with the quotient.

A duality theorem

This chapter presents a duality theorem providing, for each geometric theory, a natural bijection between its geometric theory extensions (also called ‘quotients’) and the subtoposes of its classifying topos. Two different proofs of this theorem are provided, one relying on the theory of classifying toposes and the other, of purely syntactic nature, based on a proof-theoretic interpretation of the notion of Grothendieck topology. Via this interpretation the theorem can be reformulated as a proof-theoretic equivalence between the classical system of geometric logic over a given geometric theory and a suitable proof system whose rules correspond to the axioms defining the notion of Grothendieck topology. The role of this duality as a means for shedding light on axiomatization problems for geometric theories is thoroughly discussed, and a deduction theorem for geometric logic is derived from it.

Lattices of theories

In this chapter, by using the duality theorem established in Chapter 3, many ideas and concepts of elementary topos theory are transferred into the context of geometric logic; these notions notably include the coHeyting algebra structure on the lattice of subtoposes of a given topos, open, closed, quasi-closed subtoposes, the dense-closed factorization of a geometric inclusion, coherent subtoposes, subtoposes with enough points, the surjection-inclusion factorization of a geometric morphism, skeletal inclusions, atoms in the lattice of subtoposes of a given topos, the Booleanization and DeMorganization of a topos. An explicit description of the Heyting operation between Grothendieck topologies on a given category and of the Grothendieck topology generated by a given collection of sieves is also obtained, as well as a number of results about the problem of ‘relativizing’ a local operator with respect to a given subtopos.

-modules on spaces of rational maps

Let$X$be an algebraic curve. We study the problem of parametrizing geometric structures over$X$which are only generically defined. For example, parametrizing generically defined maps (rational maps) from$X$to a fixed target scheme$Y$. There are three methods for constructing functors of points for such moduli problems (all originally due to Drinfeld), and we show that the resulting functors are equivalent in the fppf Grothendieck topology. As an application, we obtain three presentations for the category of$D$-modules ‘on’$B(K)\backslash G(\mathbb{A})/G(\mathbb{O})$, and we combine results about this category coming from the different presentations.