  In this paper, we study the Lusternik-Schnirelmann category of a simplicial
map between simplicial complexes, generalizing the simplicial category of a
complex to that of a map. Several properties of this new invariant are shown,
including its relevance to products and fibrations. We relate this category of
a map to the classical Lusternik-Schnirelmann category of a map between finite
topological spaces. Finally, we show how the simplicial category of a map may
be used to define and study a simplicial version of the essential category
weight of J. Strom.
