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I was pondering an answer that user Andreas Blass provided to an old question of mine and wondered if the following construction cropped up anywhere other than the example I will provide.

For any set $X$, let $\mathscr P(X)$ be the category whose objects are the subsets of $X$ and whose morphisms are the inclusion maps. Let $R\subseteq A\times B$ be a binary relation on two sets $A$ and $B$. Then there are two functors \begin{align*} F &: \mathscr P(B)\to\mathscr P(A)^{\operatorname{op}} \\ G &: \mathscr P(A)^{\operatorname{op}}\to\mathscr P(B) \end{align*} defined on objects by \begin{align*} F(X) &= \{a\in A:x\in X\Rightarrow (a,x)\in R\} \\ G(X) &= \{b\in B:x\in X\Rightarrow (x,b)\in R\} \end{align*} One need not work very hard to show that $F$ is left-adjoint to $G$.

A nice example of this occurs in the $\DeclareMathOperator{Spec}{Spec}\Spec$ construction. Let $A$ be a commutative ring and let $R\subseteq A\times \Spec A$ be $$ R=\{(a,\mathfrak p)\in A\times\Spec A: a\in\mathfrak p\} $$ The above construction provides us with the familiar functors \begin{align*} \Bbb V &:\mathscr P(A)^{\operatorname{op}}\to\mathscr P(\Spec A) \\ \Bbb I &:\mathscr P(\Spec A)\to\mathscr P(A)^{\operatorname{op}} \end{align*} defined on objects by \begin{align*} \Bbb V(X) &= \{\mathfrak p\in\Spec A:a\in X\Rightarrow (a,\mathfrak p)\in R\} \\ &= \{\mathfrak p\in\Spec A:X\subseteq \mathfrak p\} \\ \Bbb I(X) &=\{a\in A:\mathfrak p\in X\Rightarrow (a,\mathfrak p)\in R\} \\ &= \bigcap_{\mathfrak p\in X}\mathfrak p \end{align*} The abstract-nonsense above tells us that $\Bbb I\dashv \Bbb V$. Since products and coproducts in categories of the form $\mathscr P(X)$ are intersections and unions respectively, the continuity properties of adjoint-pairs give us the familiar formula $$ \Bbb V\left(\bigcup_{i\in I}X_i\right)=\bigcap_{i\in I}\Bbb V(X_i) $$ Now, I'm not claiming there's anything new or deep going on here but I am curious if this crops up anywhere else. Are there other adjunctions that arise from this construction?

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Brian Fitzpatrick
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1 Answers1

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Yes, this is a nice observation.

The adjunction between syntax and semantics is another example. In particular: Let $A$ denote a first-order language and $B$ denote the class of models for that language below some inaccessible $\kappa$. Let $R \subseteq A \times B$ denote the set of all pairs $(s,m)$ such that $s$ is a sentence satisfied by $m$. Then the functions $G$ and $F$ in your question satisfy: $G(S)$ is the set of all models satisfying every sentence in $S$, and $F(M)$ is the set of all sentences satisfied by every model in $M$. Also $F(G(S))$ is the closure of $S$ with respect to the sentences that semantically follow from $S$.

Remark. Note that this isn't really category theory, rather its the special case called order theory. Order theorists tend to call these things "Galois connections" rather than adjunctions, which explains why it was hard to find information elsewhere; your construction is described here at the Galois connection page at wikipedia.

goblin GONE
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