Category of continuous self maps

Question

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?

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? — Sergei Akbarov, Commented Mar 31, 2019 at 16:14
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 $b$ $a=ab$. — Tom Goodwillie, Commented Mar 31, 2019 at 16:32
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). — Pierre PC, Commented Mar 31, 2019 at 23:00

Jiří Rosický · Accepted Answer · 2019-04-01 14:42:55Z

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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.

answered Apr 1, 2019 at 14:42

Jiří Rosický

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