Category theoretic definition of topological spaces.

We know that for algebraic structures like groups and rings, the axioms in their definition can be written in terms of objects and morphisms in the category of sets and hence can be generalised to give group objects and ring objects given here and here respectively. I was wondering if we could do a similar thing for topological spaces.

In defining topology on a set $X$, we first identify a collection of subsets of $X$ as the open sets. Correspondingly, we have the notion of subobjects of a given object in any category given by monomorphisms $A\hookrightarrow X$ upto isomorphisms of such diagrams. Then we just have to translate the statements that

1. $X$ and the empty set $\phi$ belong to the collection. This can be given by asking the identity on X and the unique map from the initial object $*$ to $X$, in a monomorphism and is in our collection.
2. The collection is closed under finite intersection. We can translate this as for any two morphisms $A\hookrightarrow X$ and $B\hookrightarrow X$ in the collections, the pullback of the diagram $A\hookrightarrow X \hookleftarrow B$ exisits and is part of the collection.
3. The collection is closed under arbitrary unions. This is the part I find hardest to translate into a statement in category theory. What I was thinking is, given a collection of maps $A_i\hookrightarrow X$, we have the induced map from the coproduct, $\coprod_iA_i\to X$. The problem is, this map is not a monomorphism in general, so to get a monomorphism I think we need some kind of canonical epi-mono factorisation.

My question is can these arguments be completed to get a category theoretic definition of topological spaces ? Is this construction useful in anyway or does it have serious drawbacks?

• I haven't really looked into what you did exactly, but here is a thought: The common definition of a topological space is quite "classical". Usually, if you want to internalize a construct in a category, it needs to make sense in constructive mathematics. This is the case for groups and rings (and every other kind of universal algebra), but not for topological spaces (and e.g. not for fields either). An alternative to topological spaces are locales: ncatlab.org/nlab/show/locale . Trying to internalize those may be a more fruitful endeavor. – Stefan Perko Apr 9 '16 at 7:15
• @StefanPerko Could you elaborate on what you mean by "make sense in constructive mathematics". – Arun Kumar Apr 9 '16 at 7:26
• It's hard to make this precise, but: First of all you need a single constructive definition (e.g. the definition of a field breaks down in many different definitions constructively) and additionally certain theorems you expect to hold, should hold. Consider for example, that you cannot constructively prove, that the interval $[0,1]$ as a topological space is compact, however the locale $[0,1]$ is compact constructively. If you are wondering why all this business about constructive notions is important: It's because the internal logic of a topos is constructive, but "rarely" classical. – Stefan Perko Apr 9 '16 at 7:39
• So a category with some limits and colimits has some kind of weakened internal constructive logic (which is rarely classical). – Stefan Perko Apr 9 '16 at 7:41
• @StefanPerko If our definition breaks into several different definitions than doesn't it still make sense to deal with these definitions and see if some of them are useful? I am curious because the problem with the locales view is that we cannot talk about spaces where you have topologically indistingushible points. Also could you tell me which axioms of topological spaces are more likely to prove problematic? – Arun Kumar Apr 9 '16 at 9:24

The most useful such axiomatic is the notion of Grothendieck topology, which abstracts the notion of open covering instead of open subset. (In other contexts, it's generally too strong to assume "open subsets" are subobjects.) You can also realize topological spaces as relational algebras for the ultra filter monad, whose actual algebras are compact Hausdorff spaces. But your notion does encapsulate the classical definition-just use unions of subobjects, i.e. coproducts in the category of subobjects of $X$, to get legitimate unions instead of disjoint ones.