  Acyclic categories were introduced by Kozlov and can be viewed as generalised
posets. Similar to posets, one can define their incidence algebras and a
related topological complex. We consider the incidence algebra of either a
poset or acyclic category as the quotient of a path algebra by the parallel
ideal. We show that this ideal has a quadratic Groeobner basis with a
lexicographic monomial order if and only if the poset or acyclic category is
lex-shellable.
