Hom-polytopes
Hom-polytopes
摘要：
研究了两个多面体之间仿射映射的多面体--hom-polytopes。hom-polytope函子有一个左伴随--张量积多面体。与向量空间范畴的类比是有限的，正如我们通过一系列表现出各种极值性质的明确例子所说明的那样。hom-polytopes的主要挑战是确定它们的顶点。秩零定理的一个多面体类似物相当于理解顶点映射相对于它们的满射因子和内射因子的行为。这导致有趣的类满射映射。在最后两个部分中，我们关注两种相反的极端情况--当源和目标多面体都是多边形并且是通用的或规则的。
英文摘要：
We study the polytopes of affine maps between two polytopes -- the hom-polytopes. The hom-polytope functor has a left adjoint -- tensor 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.
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