The Hopf monoid of matroids is an algebraic object that understands the geometry of the matroid polytope in a beautiful way. We consider an ordered version of this and show that it embeds in many different Hopf monoids that understand different aspects of matroid theory. Each such monoid contains shifted simplicial complexes. The goal of the talk is to explain the combinatorial nature of the big family of Hopf monoids and pose geometric questions motivated by the algebraic properties of the construction. The talk will be basic and assumes no previous knowledge of Hopf monoids. Based on joint work with Jeremy Martin.