Intersection property and interaction decomposition

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).…”

“…The motivation for the study of such functors is the celebrated decomposition into interaction subspaces for factor spaces (see [9]); we will recall briefly what is it now. Let us recall that E a = i∈a E i ; the projection P a from E = i∈I E i to E a induces an inclusions of functions f : E a → R in the space of functions from E to R. Such functions that depend only on x a , i.e.…”

“…As the study of decomposition into interaction subspaces for factor spaces was the reason why more general decomposition where introduced in previous work [9] [10] [13], we decide to call decomposition of decomposable functors or cofunctors as interaction decompositions.…”

Interaction decomposition for presheaves

“…To answer this question one needs to know how to decompose each U(a) in a compatible manner with the poset structure A and compatible with the orthogonal projection π a : V → U(a). The study of when one can find a decomposition of U compatible with A is the subject of [11] (injective case). In Appendix B [3], in the context of factor spaces (A = {0, 1} n see Definition 2.1) and for a specific scalar product one can show that such decomposition exists (Proposition B.4 [3]) (a similar result holds is stated in [1], [7]).…”