Profunctors between posets and Alexander duality

TL;DR

This paper explores profunctors between posets, introducing graph and ascent concepts, and demonstrates their relation to Alexander duality, with applications in algebraic combinatorics and topology.

Contribution

It introduces a new framework linking profunctors, cuts, and simplicial complexes, extending Alexander duality to poset-related algebraic structures.

Findings

Cuts in Boolean lattices correspond to simplicial complexes.

Profunctors between natural numbers relate to order-preserving maps with infinity.

A topology on profunctors is introduced for infinite posets.

Abstract

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