  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.
