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Question. Are there any references that develop general topology from the viewpoint of a functor $$\Phi : \mathbf{Rel} \rightarrow \mathbf{Rel}$$ that assigns to every set $X$ the set $\Phi(X)$ of filters on $X$? (I'm not quite sure how to view $X \mapsto \Phi(X)$ as a functor on $\mathbf{Rel}$, but I'll bet it can indeed be viewed as such.)

If the answer is "no", any attempts to do this as an answer will be thoroughly appreciated.

Anyway, here's what I have in mind:

  • A topological space should be a coalgebra for an appropriate comonad structure on $\Phi$; we think of the relevant map $X \rightarrow \Phi(X)$ as assigning to each point of $X$ its neighbourhood filter.

  • A convergence space should be a "relational algebra" for an appropriate "relational monad" structure on $\Phi$; we think of the relevant map $\Phi(X) \nrightarrow X$ as assigning to each filter its set of limit points.

Some further thoughts:

Given a function $N : A \rightarrow \Phi(B)$, we get a corresponding relation $\mathrm{lim}_N : \Phi(B) \nrightarrow A$ defined by $a \in \mathrm{lim}_N(F) \leftrightarrow N(a) \subseteq F$. I suppose it makes sense to call $N$ Hausdorff iff this relation is deterministic. And it should be straightforward to show that each topological space $X \rightarrow \Phi(X)$ induces a convergence space $\Phi(X) \nrightarrow X$, as defined above.

Let me talk about limits a bit. Suppose we're trying to make sense of the limit $\lim_{x \rightarrow 0}\frac{x}{x}.$ There's the standard Euclidean neighbourhood function on $\mathbb{R}$; lets call it $$E : \mathbb{R} \rightarrow \Phi(\mathbb{R}).$$

This induces a notion of convergence of filters, $$\mathrm{lim}_E : \Phi(\mathbb{R}) \nrightarrow \mathbb{R}.$$

And, I think it makes sense to interpret the limit of interest as

$$\lim_{x \rightarrow 0}\frac{x}{x} = \left(\mathrm{lim}_E \circ \Phi\left(x \mapsto \frac{x}{x}\right) \circ E\right)(0)$$

Or maybe we need to work with punctured neighbourhoods or something. Honestly, I've never really understood this stuff. Anyway, we should be able to interpret left and right limits similarly. For instance, let $E^+$ denote the neighbourhood function corresponding to the lower limit topology on $\mathbb{R}$. Then:

$$\lim_{x \rightarrow 0^+}\frac{x}{x} = \left(\mathrm{lim}_E \circ \Phi\left(x \mapsto \frac{x}{x}\right) \circ E\right)(0)$$

Again, I'm not quite sure about puncturedness. But you get the idea; limits are defined by conjugating the function of interest by a convergence structure on one side and a topological structure on the other, and then evaluating at the point of interest.

Following this line of thought, I think a reasonable definition of continuous functions between topological spaces $(X,N_X) \rightarrow (Y,N_Y)$ would be: $$f \subseteq \mathrm{lim}_{N_Y} \circ \Phi(f) \circ N_X,$$ or something like that. This should agree with the other possible definitions, namely, the definition that only mentions neighbourhoods, and the definition that only mentions convergence.

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Almost this approach has been initiated by Barr in "Relational algebras" (1970). Recent monograph with lots of references on the subject is "Monoidal topology" by Hofmann, Seal and Tholen.

The difference is that topological spaces are viewed as lax algebras rather than coalgebras, and ultrafilters rather than filters are used.

Browsing the nLab entry on it that Todd Trimble linked to in the comment below, I saw that I forgot to mention the famous Manes theorem that started it all -- the category of compact Hausdorff spaces is isomorphic to the category of algebras over the ultrafilter monad.

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