Interaction decomposition for presheaves

Abstract

Contents 1. Introduction 1 2. What is an interaction decomposition? 2 3. Necessary and sufficient condition for the interaction decomposition of projectors from a finite poset to Vect 6 4. Interaction Decomposition for presheaves in Mod 11 5. Conlusion 21 Notations 21 References 22Abstract. Consider a collection of vector subspaces of a given vector space and a collection of projectors on these vector spaces, can we decompose the vector space into a product of vector subspaces such that the projectors are isomorphic to projections?We provide an answer to this question by extending the relation between the intersection property and the interaction decomposition ([11]) to the projective case. This enables us to classify the decompositions of interactions for factor spaces. We then extend these results for presheaves from a poset to the category of modules by adding the data of a section functor when it exists.

Regionalized Optimization

“…In previous work we introduced the notion of decomposable functor [8] and presheaves [9] from a poset to the category of modules, and decomposable functor from a poset to the category of Hilbert spaces [10]. These decompositions are the natural framework for generalizing the celebrate decomposition into interaction subspaces [11] and play a role in a categorical formulation of statistical physics [12] (Chapter 8).…”

“…In Definition 2.2 the requirement is that F is a presheaf over A , if F where a functor on A Theorem 2.1 would not hold. To remedy to this one needs to consider a functor (G, F ) to Split(R − Vect f ), the subcategory of R − Vect × R − Vect op as defined in [10], that we will simply denote as Split. Split is the category which objects are vector spaces and morphisms between two vector spaces V 1 , V 2 , are couples of linear transformations π :…”