Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ?
Namely, is there a reasonable way to:
(1) given a finite topological space $ X$ , construct a finite simplicial set $ nX$ .
(2) given two finite topological spaces $ X$ and $ Y$ , construct a simplicial set $ Map(X,Y)$ whose geometric realisation is homotopy equivalent to the mapping space $ Map(|nX|,|nY|)$ between the geometric realizations of $ nX$ and $ nY$ .
(3) etc. (higher coherences)
Note: One can, of course, define $ Map(X,Y)$ to be the derived mapping space between $ nX$ and $ nY$ . But I’m wondering whether there’s something more along the lines of “take the (finite) set of all continuous maps $ X\to Y$ and then … “
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