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Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)?

How can we tell whether a category is the category of continuous self maps of some topological space?

Are there at least existing theorems or frameworks for questions related to these ones?

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    $\begingroup$ What exactly do you mean by category? If maps are morphisms, then wouldn't it be better to call this a monoid, not a category? $\endgroup$ Mar 31, 2019 at 16:14
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    $\begingroup$ Assuming that you mean the monoid, here are two comments: You cannot distinguish between a one-point space and the empty space in this way, and if $X$ is a nonempty space then you can describe the point set of $X$ as the set of all monoid elements $a$ such that that for every $b$ $a=ab$. $\endgroup$ Mar 31, 2019 at 16:32
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    $\begingroup$ what are morphisms between continuous maps here? $\endgroup$ Mar 31, 2019 at 16:37
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    $\begingroup$ $a = ab$ for every $b$ $\endgroup$
    – DCM
    Mar 31, 2019 at 16:44
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    $\begingroup$ If your only consider the algebraic structure of the monoid, you cannot distinguish between the discrete and trivial topologies (all "abstract" maps are continuous in both cases). $\endgroup$
    – Pierre PC
    Mar 31, 2019 at 23:00

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Following the comment of Tom Goodwillie, you can recognize the underlying set and the underlying maps from the monoid. This leads to http://matwbn.icm.edu.pl/ksiazki/fm/fm66/fm6614.pdf.

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