Profunctors between posets and Alexander duality

We consider profunctors $f:P\text{promap}Q$ between posets and introduce their {\em graph} and {\em ascent}. The profunctors $\text{Pro}(P,Q)$ form themselves a poset, and we consider a partition $\text{cI}\bigsqcup \text{cF}$ of this into a down-set $\text{cI}$ and up-set $\text{cF}$, called a {\it cut}. To elements of $\text{cF}$ we associate their graphs, and to elements of $\text{cI}$ we associate their ascents. Our basic result is that this, suitable 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 $\text{Pro}(\text{NN},\text{NN})$. Such profunctors identify as order preserving maps $f : \NN \pil \NN \cup {\infty }$\mathrm{.}Forourapplicationswhen$P$and$Q$ are infinite, we also introduce a topology on $\text{Pro}(P,Q)$, in particular on profunctors $\text{Pro}(\text{NN},\text{NN})$.