Profunctors Between Posets and Alexander Duality

Abstract

We consider profunctors between posets and introduce their graph and ascent. The profunctors \(\text {Pro}(P,Q)\) form themselves a poset, and we consider a partition \(\mathcal {I}\sqcup \mathcal {F}\) of this into a down-set \(\mathcal {I}\) and up-set \(\mathcal {F}\), called a cut. To elements of \(\mathcal {F}\) we associate their graphs, and to elements of \(\mathcal {I}\) 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 \(Q \times P\). 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 letterplace ideals for posets. We study \(\text {Pro}({\mathbb N}, {\mathbb N})\). Such profunctors identify as order preserving maps \(f: {\mathbb N}\rightarrow {\mathbb N}\cup \{\infty \}\). For our applications when P and Q are infinite, we also introduce a topology on \(\text {Pro}(P,Q)\), in particular on profunctors \(\text {Pro}({\mathbb N},{\mathbb N})\).