I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't isomorphic to the total singular complex of a CGWH space. Are there?) Would someone mind providing an example of one (and an example for the opposite statement as well, if it is true)?



Example of a CW complex not homeomorphic to the realization of a simplicial set?
The mapping cylinder of a really messy continuous map $I\to I$


The nerve of the category in which there are two objects and each Hom set is a singleton.


