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The category $\mathbf{Top}_\ast$ of pointed topological spaces can be viewed as the comma category $(\bullet\downarrow\mathbf{Top})$.

The objects of the category $\mathbf{Top}$ are topological spaces and the morphisms are the continuous maps between them. The composition of morphisms is the usual composition of maps and the identity morphisms are the identity maps.

Is there exists another way to describe the category $\mathbf{Top}$ of topological spaces with other concepts in category theory?

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    $\begingroup$ What kind of description do you have in mind? $\endgroup$ – Clive Newstead May 6 '16 at 13:13
  • $\begingroup$ @CliveNewstead I am not an expert in category theory. Maybe we can construct $\mathbf{Top}$ from $ \mathbf{Set}$ by applying some concepts in category theory. $\endgroup$ – M.A. May 6 '16 at 13:47
  • $\begingroup$ You can look in the comments to zyx's answer to this question. @Pece refers there to a categorical description of topological spaces using sketches. $\endgroup$ – Mariano Suárez-Álvarez May 6 '16 at 17:56
  • $\begingroup$ @MarianoSuárez-Alvarez [this question](???) :). $\endgroup$ – M.A. May 7 '16 at 2:59
  • $\begingroup$ @M.A., there you go :-) $\endgroup$ – Mariano Suárez-Álvarez May 7 '16 at 3:03
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Most certainly. One interesting way is to write the category of spaces as the category of relational algebras for the ultrafilter monad on sets. This is a fancy way of saying that a space is the data of a set together with the choice of a set of limit points for every ultrafilter on that set, such that principal ultrafilters converge to the point they're based at and there's a compatibility with the composition of ultrafilters on the set of ultrafilters.

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