Affine-compact functors

TL;DR

This paper explores polytopal constructions through a functorial lens, correcting previous analogies and introducing new affine-compact functors, with implications for polytope theory and stochastic factorizations.

Contribution

It provides a corrected functorial formula using the affine-compact kernel and introduces new affine-compact functors related to sandwiched simplices.

Findings

Disproved naive analogy between fiber polytope and abelian kernel.

Established a functorial formula involving the affine-compact kernel.

Connected affine-compact functors to stochastic factorizations.

Abstract

Several well known polytopal constuctions are examined from the functorial point of view. A naive analogy between the Billera-Sturmfels fiber polytope and the abelian kernel is disproved by an infinite explicit series of polytopes. A correct functorial formula is provided in terms of the affine-compact kernel. The dual cokernel object is almost always the natural affine projection. The Mond-Smith-van Straten space of sandwiched simplices, useful in stochastic factorizations, leads to a different kind of affine-compact functors and new challenges in polytope theory.