  We show that various combinatorial invariants of matroids such as Chow rings
and Orlik--Solomon algebras may be assembled into "operad-like" structures.
Specifically, one obtains several operads over a certain Feynman category which
we introduce and study in detail. In addition, we establish a Koszul-type
duality between Chow rings and Orlik--Solomon algebras, vastly generalizing a
celebrated result of Getzler. This provides a new interpretation of
combinatorial Leray models of Orlik--Solomon algebras.
