Categorical valuative invariants of polyhedra and matroids

Abstract

We introduce the notion of a categorical valuative invariant of polyhedra or matroids, in which alternating sums of numerical invariants are replaced by split exact sequences in an additive category. We provide categorical lifts of a number of valuative invariants of matroids, including the Poincare polynomial, the Chow and augmented Chow polynomials, and certain two-variable extensions of the Kazhdan--Lusztig polynomial and Z-polynomial. These lifts allow us to perform calculations equivariantly with respect to automorphism groups of matroids.