Higher categories from oriented matroids

Abstract

In the recent paper [LMP25], Laplante-Anfossi, Medina-Mardones and Padrol constructed a certain chain complex C(P) from a framed polytope P that is equivalent, via a functor first defined by Steiner in [Ste04], to an ω-category. This construction was partially motivated by the Kapranov-Voevodsky conjecture formulated in [KV91], which first attempted to define a certain ncategory, labeled a pasting diagram, from an n-dimensional framed polytope. Our goal in this TFM is to generalize the construction in [LMP25] to acyclic oriented matroids, which are purely combinatoric abstractions of geometric structures such as point configurations. Thus, we construct a chain complex C(M, β) from an acyclic oriented matroid with an associated poset orientation β such that it can itself determine an ω-category, partially due to its unitality, which we prove. For that, we define the notion of admissible flags of oriented matroids to generalize the notion of frames for polytopes, we consider the Las Vergnas face lattice F(M) of the acyclic oriented matroid Mas a generalization of the face lattice of a framed polytope, and we define on it a thin poset orientation β, which is used to define the boundary of the chain complex C(M, β). In order to fully understand this construction and its relationship to ω-categories, we also explain Steiner’s original construction [Ste04] and its reinterpretation in [LMP25], we detail the construction of C(P) in [LMP25], and introduce the relevant concepts of oriented matroid theory.