4.4. Adjoints on Preorders.

Interesting things happen when one applies adjoint concepts to functors between preorders; ones which preserve order in a special way. It's actually often the case where we have two mathematical structures involving chains of arrows which reverse when transferring between one and the other. We give such a concept a definition first, before introducing a theorem about such structures.

Let \(\pp\) and \(\qqq\) be two preorders. If there exists functors \(F: \pp \to \qqq\) and \(G:\qqq \to \pp\) such that

\[ F(P) \le Q \iff P \le G(Q), \]

That is, there exists \(f:F(P) \to Q\) if and only if there exists \(g: P \to G(Q)\), then \(F\) and \(G\) are called a monotone Galois connection. On the other hand, if we have that

\[ F(P) \le Q \iff P \ge G(Q) \]

then \(F\) and \(G\) are called a antitone Galois connection.

Let \(\mathcal{P}, \mathcal{Q}\) be two preorders, and suppose \(F: \mathcal{P} \to \mathcal{Q}\op\) and \(G:\mathcal{Q}\op \to \mathcal{P}\) are two order preserving functors. Then \(F\) is left adjoint to \(G\) if and only if for all \(P \in \mathcal{P}\) and \(Q \in \mathcal{Q}\)

\[ F(P) \ge Q \iff P \le G(Q). \]

Given such an adjunction, we then have that our unit establishes \(P \le G(F(P))\) and the counit establishes \(F(G(Q)) \le Q\).

Observe that if \(F\) is left adjoint to \(G\), then we have the bijection

\[ \hom_{\mathcal{Q}\op}(F(P), Q) \cong \hom_{\mathcal{P}}(P, G(Q) ) \]

which gives rise to the desired correspondence; on the other hand, such a bijection gives rise to an adjunction. With such an adjunction, we know that for each \(P, Q\), there exist morphisms \(\eta_P: P \to G(F(P))\) and \(\epsilon_Q: F(G(Q)) \to Q\). Hence \(P \le G(F(P))\) and \(F(G(Q)) \ge Q\).

The above theorem came out of the observation that there is a connection between fields, their subfields, and their groups of automorphisms, an observation which arises in Galois Theory. The goal of Galois Theory is to understand polynomials and their roots; when they can be factorized, when and where we can find their roots. The study of Galois groups is now used widely in number theory. For example, part of Andrew Wiles' work in proving Fermat's Last Theorem involved Galois representations.

It was this theorem, rooted in Galois Theory, that motivated the Theorem 4.\ref{galois_connections} at the beginning of this section. The Fundamental Theorem of Galois Theory is simply a stronger, special case, since in this case, the functors are literally inverses of each other. The theorem we introduced, however, simply requires the functors to be adjoints of one another.

Let \(U, V\) be sets, and observe that their power sets \(\mathcal{P} (U)\) and \(\mathcal{P}(V)\) form categories; specifically, preorders, ordered by set inclusion.

Suppose \(f: U \to V\) is a function in Set. Then \(f\) induces a functor \(f_*: \mathcal{P}(U) \to \mathcal{P}(V)\), where

\[ f_*(X) = \{f(x) \mid x \in X\}. \]

Note that if \(X\subset X'\), then \(f_*(X) \subset f_*(X')\). Hence this is an order-preserving functor. Now observe that \(f\) also induces a functor \(f^*: \mathcal{P}(V) \to \mathcal{P}(U)\) where

\[ f^*(Y) = \{x \mid f(x) \in Y\}. \]

Note that this also preserves order. In addition, we have that if \(f_*(X) \le Y\), then this holds if and only if \(f(X) \subset Y\). We then have that this holds if and only if \(X \subset f_*(Y)\), Hence we have a Galois connection, so that we may apply Theorem 4.\ref{galois_connections} to conclude that \(f_*\) is left adjoint to \(f^*\).