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 ... "



Are finite spaces a model for finite CW-complexes?
The answer to question (1) is **yes** and it follows from the following theorem by McCord:



> 
> **Theorem 1.** (i) For each finite topological space $X$ there exist a finite simplicial complex $K$ and a weak homotopy equivalence $f:|K|\to X$. (ii) For each finite simplicial complex $K$ there exist a finite topological space $X$ and a weak homotopy equivalence $f:|K|\to X$.
> 
> 
> McCord, Michael C. ["Singular homology groups and homotopy groups of finite topological spaces."](https://projecteuclid.org/euclid.dmj/1077376525) *Duke Math. J* 33.3 (1966): 465-474.
> 
> 
> 


In fact, by reading the paper we see that the constructions are fairly explicit [in the following by "order topology" on a poset I mean the topology whose open sets are those $U$ where if $x\in U$ and $y>x$, then $y\in U$):


* For (i) we just take $K$ to be the nerve of $X$, seen as a poset under the specialization order ($x<y$ iff $x\in \overline{\{y\}}$) and the map $f$ is the last vertex map (sending $t\in |K|$ to the biggest vertex of the simplex containing $t$ in its interior).
* for (ii) we just take for $X$ the poset of nondegenerate simplices of $K$ with the order topology and the map $f:|K|\to X$ is the map sending every point to the simplex in whose interior it lies.


This correspondence extends to a correspondence between Alexandrov spaces (preorders with the order topology) and general simplicial complexes.


I don't know if there is an explicit way of constructing the mapping space $\mathrm{Map}(X,Y)$ without passing through the corresponding complexes. It would be interesting also if there is a simplicial model category structure on A-spaces Quillen equivalent to the Kan model structure on simplicial sets.


The [book](https://math.uchicago.edu/%7Emay/FINITE/FINITEBOOK/FINITEBOOKCollatedDraft.pdf) by May cited in Qiaochu Yuan's comment seems to contain more information about this kind of questions (Df. 5.5.3 seems to be giving a criterion for when two maps of A-spaces are homotopic).


