Language is contextual and sheaf theory provides a high level mathematical framework to model contextuality. We show how sheaf theory can model the contextual nature of natural language and how gluing can be used to provide a global semantics for a discourse by putting together the local logical semantics of each sentence within the discourse. We introduce a presheaf structure corresponding to ...

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...

This paper is concerned with a relationship between the existence of subvarieties of principally polarized abelian varieties (ppav’s) having minimal cohomology class and the (generic) vanishing of certain sheaf cohomology, based on the Generic Vanishing criterion studied in [PP3]. This is in analogy with the well-known equivalence between a subvariety in projective space being of minimal degree...

We prove a conjecture of A. S. Buch concerning the structure constants of the Grothendieck ring of a flag variety with respect to its basis of Schubert structure sheaves. For this, we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities, have alternating signs. Equivalently, the class of the dualizing sheaf of such a subvariety is a nonne...

HomX denotes the sheaf-Hom functor of OX -complexes: HomX(E,F )(U) := Hom • U (E|U , F |U ) (U ⊂ X open), the restriction map for U ′ ⊂ U being the obvious one. This “dynamic” sheafified version of Hom• has a derived functor RHom•, defined as usual via q-injective resolutions (which always exist!). Similarly, we have a sheaf-theoretic version of ⊗, and its left-derived functor ⊗ = , defined via...

A formulation in terms of sheaf theoretic (or categorical) notions for quantum entanglement is given with direct experimental consequences. The notions from sheaf theory and category theory give structural theory, i.e., qualitative theory, as a candidate for quantum gravity. Its advantage is the following: it provides not only space-time background independent, but also scale independent.This t...

We show that the cohomology table of any coherent sheaf on projective space is a convergent—but possibly infinite—sum of positive real multiples of the cohomology tables of what we call supernatural sheaves. Introduction Let K be a field, and let F be a coherent sheaf on P = P K . The cohomology table of F is the collection of numbers γ(F) = (γi,d) with γi,d = dimH (P,F(d)), which we think of a...

In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of mv-algebras and mv-chains, to the representation of commutative rings with unit as rings of global sec...

A few years ago, I defined a squarefree module over a polynomial ring S = k[x1, . . . , xn] generalizing the Stanley-Reisner ring k[∆] = S/I∆ of a simplicial complex ∆ ⊂ 2. This notion is very useful in the StanleyReisner ring theory. In this paper, from a squarefree S-module M , we construct the k-sheaf M on an (n − 1) simplex B which is the geometric realization of 2. For example, k[∆] is (th...

Given a Heegaard splitting of a three-manifold Y , we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an int...