  In this paper we consider a family of abstract simplicial complexes which we
call immediate snapshot complexes. Their definition is motivated by theoretical
distributed computing. Specifically, these complexes appear as protocol
complexes in the general immediate snapshot execution model.
  In order to define and to analyze the immediate snapshot complexes we use the
novel language of witness structures. We develop the rigorous mathematical
theory of witness structures and use it to prove several combinatorial as well
as topological properties of the immediate snapshot complexes. In particular,
we prove that these complexes are simplicially homeomorphic to simplices.
