Is a topological space a structure?

Question

In model theory, a structure (or "model") is typically defined as a set together with some finitary relations and/or operations on that set. For instance, a group can be viewed as a pair $(G,*),$ where $*$ is a binary operation defined on $G$.

Under this definition, can a topological space be viewed as a structure? I would think "no."

Supposing not, my question is this. Short of full-blown categorification, wherein structures become "black-boxes" differentiated only by the morphisms between them, is there a more general definition of structure that encompasses topological spaces, too?

I've put "category-theory" as a tag in the hope of getting more leads.

Answer 1

You can get close if you restrict to compact Hausdorff spaces: these turn out to be precisely sets together with a family of infinitary operations, one for each ultrafilter on every index set $I$. This operation computes the limit with respect to the ultrafilter, the topology is completely determined by limits of ultrafilters, and a continuous function is precisely a function respecting limits of ultrafilters. (This definition makes sense for every topological space, except that a topological space is compact iff ultrafilters always have at least one limit and Hausdorff iff ultrafilters always have at most one limit, hence we get a genuine operation iff the space is compact Hausdorff).

Anyway, you can augment the definition of structure however you want. Topological spaces don't fit into the framework you describe because there's a second set floating around, namely the two-element set $2 = \{ 0, 1 \}$. Functions $X \to 2$ describe subsets of $X$, so a topology on $X$ is a function $(X \to 2) \to 2$ satisfying some axioms. In the end, model theory is nice and all but the categories you get from it are somewhat restricted relative to all of the other wonderful categories out there.

Answer 2

Topological spaces are not algebraic structures (in the sense of monads). Indeed, in the presence of the axiom of choice, one can show that any category that is monadic over $\textbf{Set}$ is a regular category, but $\textbf{Top}$ is known to be not regular. However, the category $\textbf{Frm}$ of frames is monadic over $\textbf{Set}$, so the theory of locales (= pointless topological spaces) is, in some sense, coalgebraic.

There is also a category-theoretic notion of topological structure that goes something like this.

Definition. Let $\Gamma : \mathcal{C} \to \mathcal{S}$ be a functor and let $B : \mathcal{J} \to \mathcal{C}$ be a (possibly large) diagram. A $\Gamma$-initial cone is a cone $\alpha$ from an object $A$ to the diagram $B$ such that, for all cones $\beta$ from $\Gamma C$ to $\Gamma B$, there exists a unique morphism $f : C \to A$ in $\mathcal{C}$ such that $\Gamma (\alpha_j \circ f) = \beta_j$ for all $j$ in $\mathcal{J}$.

A topologising fibration (or topological functor in the sense of [Adámek, Herrlich, and Streicher]) is a functor $\Gamma : \mathcal{C} \to \mathcal{S}$ such that, for every (possibly large) discrete diagram $B : \mathcal{J} \to \mathcal{C}$ and every cone $\beta$ from an object $X$ to the diagram $\Gamma B$, there exists a $\Gamma$-initial cone $\alpha$ such that $\beta = \Gamma \alpha$.

In effect, what we are axiomatising is the existence of initial topologies. This definition already allows to prove many things: for example, any topologising fibration is faithful, has both left and right adjoints, and is a Grothendieck fibration. Moreover, the definition is self-dual: $\Gamma : \mathcal{C} \to \mathcal{S}$ is a topologising fibration if and only if $\Gamma^\textrm{op} : \mathcal{C}^\textrm{op} \to \mathcal{S}^\textrm{op}$ is a topologising fibration, i.e. we also get $\Gamma$-terminal lifts of cocones!