Let $X$ be a set. We can turn $\mathcal P(X)$ (the power set of $X$) into a category by taking inclusion maps as morphisms. Now consider a function $f : X \to Y$, which induces the functor $f^{-1} : \mathcal P(Y) \to \mathcal P(X)$. Now we have the identities
$$
f^{-1} \left( \bigcup_{\alpha} V_{\alpha} \right) = \bigcup_{\alpha} f^{-1}(V_{\alpha}), \quad f^{-1} \left( \bigcap_{\alpha} V_{\alpha} \right) = \bigcap_{\alpha} f^{-1}(V_{\alpha}).
$$
A limit/colimit in the category $\mathcal P(X)$ is a intersection/union of subsets of $X$, so the first equation says that $f^{-1}$ commutes with all limits/colimits. If our categories are nice enough (which we assumed, since everything here is small, c.f. this Wikipedia page), the Special Adjoint Functor Theorem tells us that $f^{-1}$ should have a left-adjoint and a right-adjoint functor. This :
$$
f(U) \subseteq V \quad \Longleftrightarrow \quad U \subseteq f^{-1}(V)
$$
tells us that
$$
\mathrm{Hom}_Y(f(U),V)) \simeq \mathrm{Hom}_X(U, f^{-1}(V))
$$
so I assumed that $f : \mathcal P(X) \to \mathcal P(Y)$ mapping $U \mapsto f(U)$ was the left-adjoint functor I was looking for, so that $f^{-1}$ has a left-adjoint and preserves all limits, we are happy.
Now the thing is I can't find the right adjoint ; it's not $f$, and I've tried several other things, doesn't work.
Question : First of all, did I understand all the above correctly, or did I make a mistake somewhere? I am not quite acquainted with the concepts of limits/colimits and the theorem, I am still learning this stuff.
Second of all, assuming the answer to the first question is yes, what is the right-adjoint of $f^{-1}$?
Best Answer
Powersets, as posets / categories, are self-dual via taking complements. This implies that the right adjoint is the complement of the image of the complement.
A conceptual way to think about the left and right adjoints of taking inverse image is that they are given by fiberwise existential vs. universal quantification. That is, if $f : X \to Y$ is a map of sets and $f^{\ast} : 2^Y \to 2^X$ the inverse image map, its left adjoint takes a subset $S \subseteq X$ to the set
$$T = \{ y \in Y : \exists x \in f^{-1}(y) : x \in S \}$$
while its right adjoint takes a subset $S \subseteq X$ to the set
$$T = \{ y \in Y : \forall x \in f^{-1}(y) : x \in S \}.$$