  We study pure ordered simplicial complexes (i.e., simplicial complexes with a
linear order on their ground sets) from the Hopf-theoretic point of view. We
define a \textit{Hopf class} to be a family of pure ordered simplicial
complexes that give rise to a Hopf monoid under join and deletion/contraction.
The prototypical Hopf class is the family of ordered matroids. The idea of a
Hopf class allows us to give a systematic study of simplicial complexes related
to matroids, including shifted complexes, broken-circuit complexes, and
\textit{unbounded matroids} (which arise from unbounded generalized
permutohedra with 0/1 coordinates).
  We compute the antipodes in two cases: \textit{facet-initial complexes} (a
much larger class than shifted complexes) and unbounded ordered matroids. In
the latter case, we embed the Hopf monoid of ordered matroids into the Hopf
monoid of ordered generalized permutohedra, enabling us to compute the antipode
using the topological method of Aguiar and Ardila. The calculation is
complicated by the appearance of certain auxiliary simplicial complexes that we
call \textit{Scrope complexes}, whose Euler characteristics control certain
coefficients of the antipode. The resulting antipode formula is
multiplicity-free and cancellation-free.
