3
$\begingroup$

Does the functor of geometric realization of a simplicial set as a topological space, factor through an endofunctor of the category of simplicial topological spaces which does something non-trivial (in an informal sense)?

Is there a known construction showing something like this, possibly via an endofunctor of some other category? More generally, is geometric realisation known to be associated with some endofunctor ?

The motivation for the question is that it is claimed there is such a factorisation through an endofunctor of a category similar but distinct from the category of simplicial topological spaces, and I'd like to know if it is related to something.

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ You can factor the usual geometric realization functor through the category of simplicial spaces, by considering the constant simplicial space of the geometric realization. Therefore it factors through an endofunctor of the category of simplicial spaces: the identity endofunctor. This is probably not what you have in mind, but you should narrow your question to exclude such trivial answers. $\endgroup$
    – Fernando Muro
    Commented Oct 23, 2020 at 22:29
  • $\begingroup$ I do not know how to narrow it in a precise sense. Informally, it is a request for references to a non-trivial endofunctor associated with geometric realisation in some sense. $\endgroup$
    – user167192
    Commented Oct 24, 2020 at 8:41
  • $\begingroup$ You can just consider geometric realisation of simplicial spaces as an endofunctor of simplicial spaces: It takes a simplicial space to the constant simplicial space given by the geometric realisation. Is this what you had in mind? $\endgroup$
    – Achim Krause
    Commented Oct 24, 2020 at 9:19

0

Reset to default

You must log in to answer this question.