Profunctors Between Posets and Alexander Duality

Abstract

We consider profunctors f : P |−→Q between posets and introduce their graph and ascent. The profunctors Pro(P, Q) form themselves a poset, and we consider a partition I F of this into a down-set I and up-set F, called a cut. To elements of F we associate their graphs, and to elements of I we associate their ascents. Our basic results is that this, suitably refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of Q × P. Cuts in finite Booleans lattices correspond precisely to finite simplicial complexes. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letter- place ideals for posets. We study Pro(N, N). Such profunctors identify as order preserving maps f : N → N ∪ {∞}. For our applications when P and Q are infinite, we also introduce a topology on Pro(P, Q), in particular on profunctors Pro(N, N).