Simplicial fibrations

We undertake a systematic study of fibrations in the setting of abstract simplicial complexes, where the concept of "homotopy" has been replaced by that of "contiguity". Then, a fibration will be a simplicial map satisfying the "contiguity lifting property". This definition turns out to be equivalent to that introduced by Minian, established in terms of a cylinder construction KxIm. This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration with connected base have the same strong homotopy type and any fibration with a strongly collapsible base is fibrewise trivial. We also introduce the concept of "simplicial finite-fibration", that is, a simplicial map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration PK -> KxK is a finite-fibration, where PK is the simplicial complex of Moore paths introduced by Grandis. This result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we prove a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber. Finally, we introduce a definition of "Svarc genus" of a simplicial map and we are able to compare the Svarc genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers.