Algebra/Topology Seminar

Speaker: Guillaume Laplante-Anfossi

Title: Framed polytopes and higher categories

Abstract: An omega-category is a category where, in addition to (1-)morphisms between objects, there are (n+1)-morphisms between n-morphisms, for any positive integer n. In 1991, M. Kapranov and V. Voevodsky conjectured that a polytope together with a generic basis forms a loop-free omega-category, where the n-morphisms are given by the n-dimensional faces of the polytope. In this talk, I will explain why this conjecture is true up to dimension 3, and fails for polytopes of dimension 4 and higher. The situation is as bad as it can be: there are polytopes for which any choice of basis induces a loop, and the canonical basis induces loops on high dimensional random simplices with high probability. Further, omega-category structures on polytopes are universal in the sense of Mnëv, i.e. they can behave « as badly as any semi-algebraic set », obeying what R. Vakil called Murphy’s law in algebraic geometry. This is joint work with A. Padrol and A. Medina-Mardones.