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?



Category of continuous self maps
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>.


