Profunctors Between Posets and Alexander Duality

Abstract

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