  We give combinatorial models for the homotopy type of complements of elliptic
arrangements (i.e., certain sets of abelian subvarieties in a product of
elliptic curves). We give a presentation of the fundamental group of such
spaces and, as an application, we treat the case of ordered configuration
spaces of elliptic curves.
  Our models are finite polyhedral CW complexes, and our combinatorial tools of
choice are acyclic categories (small categories without loops). As a stepping
stone, we give a characterization of which acyclic categories arise as face
categories of polyhedral CW complexes.
