  We show that Schmitt's restriction species (such as graphs, matroids, posets,
etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and
that their associated coalgebras are an instance of the general construction of
incidence coalgebras of decomposition spaces. We introduce the notion of
directed restriction species that subsume Schmitt's restriction species and
also induce decomposition spaces. Whereas ordinary restriction species are
presheaves on the category of finite sets and injections, directed restriction
species are presheaves on the category of finite posets and convex maps. We
also introduce the notion of monoidal (directed) restriction species, which
induce monoidal decomposition spaces and hence bialgebras, most often Hopf
algebras. Examples of this notion include rooted forests, directed graphs,
posets, double posets, and many related structures. A prominent instance of a
resulting incidence bialgebra is the Butcher-Connes-Kreimer Hopf algebra of
rooted trees. Both ordinary and directed restriction species are shown to be
examples of a construction of decomposition spaces from certain cocartesian
fibrations over the category of finite ordinals that are also cartesian over
convex maps. The proofs rely on some beautiful simplicial combinatorics, where
the notion of convexity plays a key role. The methods developed are of
independent interest as techniques for constructing decomposition spaces.
