ion, Composition and Contracts: A Sheaf Theoretic Approach∗ Alberto Speranzon† David I. Spivak‡ Srivatsan Varadarajan† February 12, 2018 Abstract Complex systems of systems (SoS) are characterized by multiple interconnected subsystems. Typically, each subsystem is designed and analyzed using methodologies and formalisms that are specific to the particular subsystem model of computation consider...

The aim of this chapter is to construct new foundations for quantum logic and quantum spaces. This is accomplished by merging algebraic quantum theory and topos theory (encompassing the theory of locales or frames, of which toposes in a sense form the ultimate generalization). In a nutshell, the relation between these fields is as follows. First, our mathematical interpretation of Bohr’s ‘doctr...

The concept of fibring is extended to higher-order logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the meta-theorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich logics. This result is extended to weaker logics in the cases where fibring preserves conservativ...

The dual of the category of pointed objects of a topos is semiabelian, thus is provided with a notion of semi-direct product and a corresponding notion of action. In this paper, we study various conditions for representability of these actions. First, we show this to be equivalent to the existence of initial normal covers in the category of pointed objects of the topos. For Grothendieck toposes...

Eliasson, J. 2003: Ultrasheaves. Uppsala dissertations in Mathematics 30. 54 pp. Uppsala. ISBN 91-506-1716-8. This thesis treats ultrasheaves, sheaves on the category of ultrafilters. In the classical theory of ultrapowers, you start with an ultrafilter (I,U) and, given a structure S, you construct the ultrapower S/U . The fundamental result is ! Loś’s theorem for ultrapowers giving the connect...

In this paper we introduce the notion of an extensive 2-category, to be thought of as a “2-category of generalized spaces”. We consider an extensive 2-category K equipped with a binary-product-preserving pseudofunctor C : K op → CAT, which we think of as specifying the “coverings” of our generalized spaces. We prove, in this context, a van Kampen theorem which generalizes and refines one of Bro...

This talk is based on [2]. We establish, generalizing Di Nola and Lettieri’s categorical equivalence [3], a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MValgebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classi...

A quantum causal topology is presented. This is modeled after a non-commutative scheme type of theory for the curved finitary spacetime sheaves of the non-abelian incidence Rota algebras that represent ‘gravitational quantum causal sets’. The finitary spacetime primitive algebra scheme structures for quantum causal sets proposed here are interpreted as the kinematics of a curved and reticular l...

Using topos theory I prove that reasoning about probabilities can be formalized with only one simple assumption: given two sets of measures A,B, if B ⊆ A, then B is less imprecise than A.

This paper provides a partial solution to the completeness problem for Joyal’s axiomatization of open and etale maps, under the additional assumption that a collection axiom (related to the set-theoretical axiom with the same name) holds. In many categories of geometric objects, there are natural classes of open maps, of proper maps and of etale maps. Some of the forma1 properties of these clas...