We show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the “Arithmetic Site”. This site involves the tropical semiring N̄ viewed as a sheaf on the topos N̂× dual to the multiplicative semigroup of positive integers. We realize the Frobenius correspondences in the square of the “Arithmetic Site”. Résumé Le “Site Arithmétique” est l’...

Grothendieck Toposes are often considered as generalized spaces; indeed, every space gives rise to a topos of sheaves, and various invariants and constructions from (algebraic) topology can be generalized to the level of toposes. In this talk, I will introduce a newly discovered invariant called the isotropy group of a topos and illustrate by considering special cases such as continuous group a...

The unity of opposites in the title is essentially that between logic and geometry, and there are compelling reasons for maintaining that geometry is the leading aspect. At the same lime, in the present joint work with Myles Tierney there are important influences in the other direction: a Grothendieck " topology " appears most naturally as a modal operator, of the nature " it is locally the cas...

Canonical extension has proven to be a powerful tool in algebraic study of propositional logics. In this paper we describe a generalisation of the theory of canonical extension to the setting of first order logic. We define a notion of canonical extension for coherent categories. These are the categorical analogues of distributive lattices and they provide categorical semantics for coherent log...

We introduce various notions of partial topos, i.e. “topos without terminal object”. The strongest one, called local topos, is motivated by the key examples of finite trees and sheaves with compact support. Local toposes satisfy all the usual exactness properties of toposes but are neither cartesian closed nor have a subobject classifier. Examples for the weaker notions are local homeomorphisms...

Moggi and Hyland showed how to model various polymorphic A-calculi inside the effective topos and other rea~zability toposes; types are modelled by the so-called modest sets, which form an internal category Mod in the topos that is, in a certain sense, complete. Polymorphic types are modelled as products indexed by the object of modest sets. The same idea lets us model polymorphism in reflectiv...

We show that dependent sums and dependent products of continuous parametrizations on domains with dense, codense, and natural totalities agree with dependent sums and dependent products in equilogical spaces, and thus also in the realizability topos RT(Pω).

In the framework of the topos approach to quantum mechanics we give a representation of physical properties in terms of modal operators on Heyting algebras. It allows us to introduce a classical type study of the mentioned properties.

Abstract In times of ‘grand challenges’, design theorists dealing with complex systems are facing a dilemma: grand challenges require rule breaking, but they also the preservation, as much possible, existing resources, systems, know-how and societal values. Design for transition calls not ‘creative destruction’, preservation’. How do we model process that involves preservation’? Today, it is re...

We present a topos-theoretic interpretation of (a categorical generalization of) Fraïssé's construction in Model Theory, with applications to homogeneous models and countably categorical theories. Mathematics Subject Classi cation (2010). Primary 03G30; Secondary 03C35, 18C10, 18B25.