  We show that every involutive Hopf monoid in a complete and finitely
cocomplete symmetric monoidal category gives rise to invariants of oriented
surfaces defined in terms of ribbon graphs. For every ribbon graph this yields
an object in the category, defined up to isomorphism, that depends only on the
homeomorphism class of the associated surface. This object is constructed via
(co)equalisers and images and equipped with a mapping class group action. It
can be viewed as a categorical generalisation of the ground state of Kitaev's
quantum double model or of a representation variety for a surface. We apply the
construction to group objects in cartesian monoidal categories, in particular
to simplicial groups as group objects in SSet and to crossed modules as group
objects in Cat. The former yields a simplicial set consisting of representation
varieties, the latter a groupoid whose sets of objects and morphisms are
obtained from representation varieties.
