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The approach of category theory to the description of mathematical structures, is to look at how a class of mathematical structures relate to each other, and to forget the structure itself.

e.g. in the category of topologies Top, objects are topological spaces, and morphisms are continuous functions. From a categorical perspective, Top only contains this information about the continuous functions, and forgets the "internal structure" of each object, i.e. the topological spaces.

I have read that it is an interesting property of category theory that this relational information "captures" the information about topological spaces.

Does this mean that we have literally all information about topological spaces in its category? e.g. can we rederive the axioms of topology from purely the categorical information in Top?

If not, then what does it mean concretely to say that the category $\textbf{Top}$ "captures what a topology is"?

user56834
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Let $P$ be the one point topological space, and let $U=\{u_1,u_2\}$ be the two-point topological space with open sets $\{\}$, $\{u_1\}$, and $\{u_1,u_2\}$. Then if $X$ is an arbitrary topological space, the set of points of $X$ is identified with $\mathrm{Hom}(P,X)$, and the set of open sets of $X$ is identified with $\mathrm{Hom}(X,U)$ (if $f:X\to U$ is continuous, then $f^{-1}(u_1)$ is an open subset of $X$, and every open subset comes from a unique $f$).

So we can "see" the internal structure of a topological space from just the categorical structure. With some care, every topological property can be phrased in terms of continuous maps. For example, a subspace of $X$ is an (isomorphism class of a) continuous map $f:Y\to X$ such that $\mathrm{Hom}(P,Y)\to \mathrm{Hom}(P,X)$ is injective and $\mathrm{Hom}(X,U)\to\mathrm{Hom}(Y,U)$ is surjective.

Julian Rosen
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    Note that moreover, $P$ and $U$ can be defined purely categorically. $P$ is a terminal object, and $U$ is an object with two morphisms from $P$ and no nontrivial automorphisms. The two morphisms $P\to U$ can also be distinguished categorically though this is more complicated (for instance, you can distinguish them using infinite products, essentially because infinite intersections of closed sets are closed but the same is not true for open sets). – Eric Wofsey Feb 09 '19 at 16:02
  • @EricWofsey The morphism $P\to U$ that maps to $u_1$ is also simply the unique morphism of the form $f\circ g$ where $g\colon P\to U$, $id\ne f\colon U\to U$ – Hagen von Eitzen Feb 09 '19 at 19:18
  • @HagenvonEitzen: That's not true, since $f$ could be either of the two constant maps. More generally, you cannot distinguish $u_1$ and $u_2$ using only finite topological spaces, since there is an automorphism of the category of finite spaces that swaps open and closed sets. – Eric Wofsey Feb 09 '19 at 19:32
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    Thanks, this is enlightening! Is it correct to say that when people say something like "the categorical information in Top captures the nature of topologies", this is what they mean? it seems to me that the maps to these two objects are a very small part of the structure of Top. – user56834 Feb 09 '19 at 19:47
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    Your topological space $U$ is commonly referred to as the Sierpiński space (and notated $\Sigma$), and the property you mention is often described as it being an open subspace classifier. – Derek Elkins left SE Feb 09 '19 at 19:49
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    @user56834 In any category, knowing how an object $X$ maps to every other object (really, knowing the functor $\mathrm{Hom}(X,-)$) determines $X$ up to isomorphism (this is the Yoneda lemma). The same is true if you know how every other object maps to $X$. So for abstract reasons, all of the topology of a space is captured by categorical information. In Top it is enough to know the maps from $P$ into $X$ and from $X$ into $U$, but I wouldn't assume that someone who made the remark in your comment would necessarily have $P$ and $U$ in mind. – Julian Rosen Feb 10 '19 at 00:13
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    @julianRosen, I’m confused about your mention of the yoneda lemma. As far as I understand, the yoneda lemma/embedding still doesn’t look inside the objects (i.e. topological space), so I don’t see how you can use it to actually pinpoint topological structure. My intuition is the following: The fact that you can pinpoint the topological space (the object) uniquely up to isomorphism within the category Top, doesn’t mean that you can actually deduce the topological axioms from it. – user56834 Feb 10 '19 at 05:04
  • @JulianRosen, could you edit your answer to incorporate the point about the Yoneda lemma? Then I will accept it. – user56834 Mar 25 '19 at 11:45
  • @user56834 Sure. Things are very busy at the moment, but I'll edit when I get the chance. – Julian Rosen Mar 25 '19 at 14:57