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
- $X$ 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.
- 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.
- 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?