  The goal of this paper is to show that valuation theory and Hopf theory are
compatible on the class of generalized permutahedra. We prove that the Hopf
structure $\mathbf{GP}^+$ on these polyhedra descends, modulo the
inclusion-exclusion relations, to an indicator Hopf monoid
$\mathbb{I}(\mathbf{GP}^+)$ of generalized permutahedra that is isomorphic to
the Hopf monoid of weighted ordered set partitions. This quotient Hopf monoid
$\mathbb{I}(\mathbf{GP}^+)$ is cofree. It is the terminal object in the
category of Hopf monoids with polynomial characters; this partially explains
the ubiquity of generalized permutahedra in the theory of Hopf monoids.
  This Hopf theoretic framework offers a simple, unified explanation for many
new and old valuations on generalized permutahedra and their subfamilies.
Examples include, for matroids: the Chern-Schwartz-MacPherson cycles, Eur's
volume polynomial, the Kazhdan-Lusztig polynomial, the motivic zeta function,
and the Derksen-Fink invariant; for posets: the order polynomial, Poincar\'e
polynomial, and poset Tutte polynomial; for generalized permutahedra: the
universal Tutte character and the corresponding class in the Chow ring of the
permutahedral variety.
  We obtain several algebraic and combinatorial corollaries; for example: the
existence of the valuative character group of $\mathbf{GP}^+$, and the
indecomposability of a nestohedron into smaller nestohedra.
