Sheaf representation for topoi

Sheaf representation for topoi

It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of so-called hyperlocal topoi, improving on a result of Lambek & Moerdijk. It follows that every boolean topos is equivalent to the global sections of a sheaf of well-pointed topoi. Completeness theorems for higher-order logic result as corollaries. The main result of this paper is the following. Th...

Sheaf-Theoretic Representation of Quantum Measure Algebras

We construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algeb...

Topological representation of sheaf cohomology of sites

For a site S (with enough points), we construct a topological space X(S) and a full embedding φ of the category of sheaves on S into those on X(S) (i.e., a morphism of toposes φ: Sh(X(S)) → Sh(S)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H(S, A) = H∗(X(S), φ A) for any abelian sheaf A on S. As a particular case, this will give for any sch...

Continuity and Logical Completeness: An Application of Sheaf Theory and Topoi

The notion of a continuously variable quantity can be regarded as a generalization of that of a particular (constant) quantity, and the properties of such quantities are then akin to, and derived from, the properties of constants. For example, the continuous, real-valued functions on a topological space behave like the eld of real numbers in many ways, but instead form a ring. Topos theory perm...

A Sheaf-theoretic/bayesian Approach for Shape Representation