Are finite spaces a model for finite CW-complexes?

Question

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 ... "

Answer 1

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." 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 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).

Answer 2

Andre, the best answer to your very first question is given by Emily Clader, who proved that every finite simplicial complex is weak homotopy equivalent to an inverse limit of finite spaces. A small mistake is corrected and much further work is done in Matthew Thibault's unpublished 2013 University of Chicago thesis.

The answer to your question (1) is classical, going back to McCord as in Nardin's answer. I don't know a really good answer to (2).

I should apologize that my book referred to by Quaochu Yuan is still unfinished. It will be some day. It uses the finite space of continuous maps between finite spaces to discuss homotopies in Section 2.2, but of course that is too small to realize properly. The generalization of this to A-spaces (T_0 Alexandroff spaces) is subtle and is studied by Kukiela, but he does not address your question (2).

In answer to a question raised in Nardin's answer, the category of A-spaces is isomorphic to the category of posets. It was implicit in Thomason's model structure on the category of small categories that there is a similar model structure on the category of posets, and that was made explicit by Raptis. It is Quillen equivalent to the standard model structure on simplicial sets. That was generalized to posets with action by a discrete group G by Stephan, Zakharevich and myself. In passing, that paper somewhat streamlines the nonequivariant proof.

Answer 3

An appendix to Denis Nardin's answer:

in the wonderful paper "Graduation and dimension in locales" by Isbell (in "Aspects of Topology", London MS Lecture Notes 93 (1985): 195-210), the proof of 1.4 in particular contains the following (on page 203): for a finite space $X$ define its barycentric subdivision $bX$ to be the set of those subsets of $X$ which are chains under the specialization order. This is a poset under subset inclusion order and can be viewed as another finite space (with Alexandroff topology). There is a continuous map $bX\to X$ sending a chain to its greatest element. Iterate this and consider the limit$$b^\omega X=\varprojlim\left(\ \cdots\to b^2X\to bX\to X\ \right).$$ Then the subspace of closed points of $b^\omega X$ is homeomorphic to the geometric realization of the nerve of $X$.

This suggests that maybe sensible mapping spaces can be obtained from inverse systems $\operatorname{Map}(b^iX,b^jY)$