  We extend the notion of graph homomorphism to cellularly embedded graphs
(maps) by designing operations on vertices and edges that respect the surface
topology; we thus obtain the first definition of map homomorphism that
preserves both the combinatorial structure (as a graph homomorphism) and the
topological structure of the surface (in particular, orientability and genus).
Notions such as the core of a graph and the homomorphism order on cores are
then extended to maps. We also develop a purely combinatorial framework for
various topological features of a map such as the contractibility of closed
walks, which in particular allows us to characterize map cores. We then show
that the poset of map cores ordered by the existence of a homomorphism is
connected and, in contrast to graph homomorphisms, does not contain any dense
interval (so it is not universal for countable posets). Finally, we give
examples of a pair of cores with an infinite number of cores between them, an
infinite chain of gaps, and arbitrarily large antichains with a common
homomorphic image.
