Why the preimage rather than image in Stone-type dualities.

Question

I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.

Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ generates two operators $pre: \wp(B) \to \wp(A)$ and $post: \wp(A) \to \wp(B)$ where $pre$ maps a set $X \subseteq A$ to its preimage with respect to $R$ and $post$ maps $X$ to its image.

In the standard Stone duality between the category of sets and Boolean algebras, a function is represented using the preimage operator. The preimage operator generated by a function turns out to be a Boolean algebra homomorphism but the image operator may not be a homomorphism. I see this as one reason to choose the preimage to represent a function. My first question is: Are there other reasons to choose the preimage representation? I feel like there should be something deeper going on.

If we leave the setting of functions, the preimage operator generated by a relation isn't necessarily a Boolean algebra homomorphism. So, in the representation of a system of relations over a set by a Boolean algebra with operators (in the sense of Jonsson and Tarski) I see no specific motivation for using the preimage, as opposed to image operator. I see why we want to be consistent in convention, and also use the preimage because of its connection to the semantics of modal logic. However, this appears to be an aesthetic choice. My second question is: Is there a specific reason to choose the preimage, rather than image operator, when representing relations as Boolean algebras with operators?

Answer 1

You should think of the preimage as taking the pullback of $\mathbb{F}_2$-valued functions. For any topological space $X$, the space of continuous functions $X \to \mathbb{F}_2$ may be identified with the Boolean algebra / ring of clopen subsets of $X$, the pullback of such a function is another such function, and taking pullbacks corresponds to taking preimages (thinking of subsets via their indicator functions). The natural thing to do with functions is pull them back, both in logic and in geometry. Moreover, pulling back functions is a ring homomorphism, and even in contexts where pushing forward functions can also be done it usually does not preserve multiplication.

Edit: As for relations instead of sets, taking the transpose of a relation gives an equivalence of categories $\text{Rel} \cong \text{Rel}^{op}$ ($\text{Rel}$ is a dagger category) which exchanges inverse image and image, so in that sense there isn't a strong reason to prefer one over the other for relations.

Answer 2

For a function $f:X\to Y$, the operation "pre-image along $f$" from $\mathcal P(Y)$ to $\mathcal P(X)$ has adjoints on both sides. [I'm viewing the power sets as partially ordered by $\subseteq$ and then viewing these partially ordered sets as categories, so that "adjoint" makes sense.] The left adjoint is the familiar image operator. The right adjoint is its (not so familiar) dual, $A\mapsto Y-f(X-A)$. So from this point of view, "image" has the same level of naturality as its dual, while "pre-image" is "better".

Concerning the claim of equal naturality, note that the left adjoint sends $A$ to $\{y\in Y:(\exists x\text{ with }f(x)=y)\ x\in A\}$ while the right adjoint sends $A$ to $\{y\in Y:(\forall x\text{ with }f(x)=y)\ x\in A\}$. So they correspond to the two familiar (mutually dual) quantifiers.

Answer 3

Having a subset of a set is "the same" as having a function from that set into $\{0,1\}$ (namely, the function which is $1$ on the subset and $0$ on its complement). If I have a function $f: X \to Y$ I can compose it with functions from $Y$ to $\{0,1\}$ and thereby turn subsets of $Y$ into subsets of $X$. I guess that's one explanation for why pre is natural in a way that post is not.