An acyclic category (also called loopfree category or scwol (small category without loops)) is a small category where only identity morphisms have inverses, and any morphism from an object to itself is the identity.

Every poset P can be regarded as an acyclic category by identifying the set of objects with the elements of P. And saying there is a morphism from x to y, if and only if $x\le y$. Hence we can regard acyclic categories as a generalization of posets.

Posets play a crucial role in combinatorial algebraic topology, e.g. in form of intersection lattices related to hyperplane arrangements, or face posets of simplicial complexes. I'm looking for examples where we get acyclic categories as generalized posets encoding information about some structure (e.g.: Is there some kind of generalized hyperplane arrangement which yields an intersection category).

I'm aware of the face category of a polytopal complex and the salvetti categori of a complex, toric arrangement, but I'd be glad if anyone knew some further applications for acyclic categories in the field of algebraic topology