  We show that the toric variety of the permutohedron (=permutohedral space)
has the structure of a cocommutative bimonoid in species, with
multiplication/comultiplication given by embedding/projecting-onto boundary
divisors. In terms of Losev-Manin's description of permutohedral space as a
moduli space, multiplication is concatenation of strings of Riemann spheres and
comultiplication is forgetting marked points. In this way, the bimonoid
structure is an analog of the cyclic operad structure on the moduli space of
genus zero marked curves. Covariant/contravariant data on permutohedral space
is endowed with the structure of cocommutative/commutative bimonoids by
pushing-forward/pulling-back data along the (co)multiplication. Many well-known
combinatorial objects index data on permutohedral space. Moreover,
combinatorial objects often have the structure of bimonoids, with
multiplication/comultiplication given by merging/restricting objects in some
way. We prove that the bimonoid structure enjoyed by these indexing
combinatorial objects coincides with that induced by the bimonoid structure of
permutohedral space. Thus, permutohedral space may be viewed as a fundamental
underlying object which geometrically interprets many combinatorial Hopf
algebras. Aguiar-Mahajan have shown that classical combinatorial Hopf theory is
based on the braid hyperplane arrangement in a crucial way. This paper aims to
similarly establish permutohedral space as a central object, providing an even
more unified perspective. The main motivation for this work concerns Feynman
amplitudes in the Schwinger parametrization, which become integrals over
permutohedral space if one blows-up everything in the resolution of
singularities. Then the Hopf algebra structure of Feynman graphs, first
appearing in the work of Connes-Kreimer, coincides with that induced by the
bimonoid structure of permutohedral space.
