Hom-polytopes

Abstract

We study the polytopes of affine maps between two polytopes -the hom-polytopes. The hom-polytope functor has a left adjointtensor product polytopes. The analogy with the category of vector spaces is limited, as we illustrate by a series of explicit examples exhibiting various extremal properties. The main challenge for hom-polytopes is to determine their vertices. A polytopal analogue of the rank-nullity theorem amounts to understanding how the vertex maps behave relative to their surjective and injective factors. This leads to interesting classes of surjective maps. In the last two sections we focus on two opposite extremal cases -when the source and target polytopes are both polygons and are either generic or regular.

The Inflation Technique for Causal Inference with Latent Variables

“…Since the last four tensor factors occur in every direct summand in exactly the same way, we can also write such a polytope vertex as [(e A1 ⊗ e B1 ) ⊕ (e A1 ⊗ e B2 ) ⊕ (e A2 ⊗ e B1 ) ⊕ (e A2 ⊗ e B2 )] ⊗ [e X1 ⊗ e X2 ⊗ e Y1 ⊗ e Y2 ] in ⊕ 4 i=1 R 2 2 ⊗ R 2 4 . Now since the first four variables in the first tensor factor vary completely independently of the latter four variables in the second tensor factor, the resulting polytope will be precisely the tensor product [123,124] of two polytopes: first, the convex hull of all points of the form (e A1 ⊗ e B1 ) ⊕ (e A1 ⊗ e B2 ) ⊕ (e A2 ⊗ e B1 ) ⊕ (e A2 ⊗ e B2 ), and second the convex hull of all e X1 ⊗ e X2 ⊗ e Y1 ⊗ e Y2 . While the latter polytope is just the standard probability simplex in R 8 , the former polytope is precisely the "local polytope" or "Bell polytope" that is traditionally used in the context of Bell scenarios [20, Sec.…”

Affine-compact functors

“…The set of morphisms a → b will be denoted by Hom C (a, b), or just Hom(a, b) when there is no ambiguity. For C = Sets, Vect, Pol, or Conv, the set Hom C (a, b) is naturally an object of C. This is obvious when C is Sets or Vect; when C = Pol this observation is the starting point of [5]; the case C = Conv is shown as follows: when a = ∆ is a simplex of dimension d and Y is an arbitrary convex set then Hom(∆, Y ) ∼ = Y d+1 or, equivalently, any map from the vertices of ∆ to Y uniquely extends to an affine map ∆ → Y , and for general X ∈ Conv we have…”

“…This means that there is a bifunctor ⊗ : C × C → C, together with a distinguished object I and natural isomorphisms a ⊗ I ∼ = a ∼ = I ⊗ a, (a ⊗ b) ⊗ c ∼ = a ⊗ (b ⊗ c), and a ⊗ b ∼ = b ⊗ a for any objects a, b, c ∈ C, satisfying certain coherence conditions. The monoidal product in Sets is the Cartesian product, with I a singleton; in the case of Vect, the object I is the space R and ⊗ is the tensor product of vector spaces; for Pol the object I is a singleton and the tensor product of polytopes is the dehomogenization of the usual tensor product of the associated homogenization cones; see [5,Section 3]. A particular realization is…”

“…For any P, Q ∈ Pol, the facets of Hom(P, Q) and vertices of P ⊗ Q are readily described. However, determination of the vertices of Hom(P, Q) and facets of P ⊗ Q is a real challenge and partial progress in this direction is accomplished in [5,10].…”

Quasitoric Totally Normally Split Manifolds

“…The next step is to set up two different definitions of products for convex cones. The first one is the tensor product of cones (see [5]), and the second corresponds to the cartesian product of manifolds (see Corollary 2.30).…”