Who should be credited with the proposition that if $R$ is a relation on $X \times Y$, then for all subsets $S$ of $X$, $G(F(G(S)))=G(S)$, where $G(S)$ denotes the set of $y \in Y$ such that $xRy$ for all $x \in S$ and $F(T)$ denotes the set of $x \in X$ such that $xRy$ for all $y \in T$? What is the category-theoretic way of describing this situation?
That's normally called a Galois connection (because of the example where $X$ is a field, $Y$ is a group of automorphisms of $X$, and $aRg$ means that $g(a)=a$). The Wikipedia article (http://en.wikipedia.org/wiki/Galois_connection) has a number of references.
$\begingroup$ Really?? I always thought a Galois connection was an adjoint pair of functors between posets. $\endgroup$– David Roberts ♦Jan 24, 2011 at 3:26
4$\begingroup$ Ah, I see from here en.wikipedia.org/wiki/… the situation as the OP describes gives rise to a Galois connection between the powersets of $X$ and $Y$. Cool. $\endgroup$– David Roberts ♦Jan 24, 2011 at 3:28
$G(S):= \lbrace y\in Y | \forall x\in S. xRy \rbrace$
$F(T):= \lbrace x\in X | \forall y\in T. xRy \rbrace$
This Galois connection is called “polarities” in “M. Erne, J. Koslowski, A. Melton, G. E. Strecker. A Primer on Galois Connections.” and the relation-generated Galois connection in “Smith. The Galois Connection between Syntax and Semantics.”
$\begingroup$ Thanks! Is there any terminology for the closure operator $S \mapsto F(G(S))$? $\endgroup$ Jan 28, 2011 at 17:07