How much of homotopy theory can be done using only finite topological spaces?

Question

Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. McCord constructed a $T_0$ topology on $B$ with the property that the inclusion $B \to X$ is a weak homotopy equivalence. Clader showed that the geometric realization of $X$ can be recovered up to homeomorphism by iterating McCord's construction on the $n$th barycentric subdivision of $X$, taking an inverse limit, and passing to a certain quotient space. Moreover she proved that the inverse limit deformation retracts onto the quotient space and thus every finite simplicial complex is homotopy equivalent to an inverse limit of finite spaces.

This is kind of a shocking result (though the shock wears off quickly once you stop and think about it), and it leads me to wonder:

Would it be possible to dispense with the language of simplicial complexes entirely and rewrite the foundations of homotopy theory using finite topological spaces and their inverse limits?

I am mainly interested in whether or not this is possible, though you may feel free to include in your answer remarks about whether or not this is desirable. Some possible complications that come to mind:

• Many important topological spaces (e.g. classifying spaces for most groups) do not have the homotopy type of a finite simplicial complex.
• One does not directly have access to certain tools, such as a Morse theory.

Perhaps one can hope to approximate whatever is lost using finite models?

Answer 1

Vidit, thanks for the advertisement; Paul I'll answer your email shortly. As a minor point, there is a small but subtle mistake in Clader's work that is corrected in Matthew Thibault's 2013 Chicago thesis, which goes further in that direction.

I do intend to finish the advertised book, but it is too incomplete to circulate yet. There is actually a large and interesting picture that connects mainstream algbraic topology to combinatorics via finite spaces.
However, the right level of generality is $T_0$-Alexandroff spaces, $A$-spaces for short. These are topological spaces in which arbitrary rather than just finite intersections of open sets are open, and of course finite $T_0$-spaces are the obvious examples. One can in principle answer Paul's question in the affirmative, but the finiteness restriction feels artificial and the connection between $A$-spaces and simplicial complexes is far too close to ignore.

The category of $A$-spaces is isomorphic to the category of posets, $A$-spaces naturally give rise to ordered simplicial complexes (the order complex of a poset) and thus to simplicial sets, while abstract simplicial complexes naturally give rise to $A$-spaces (the face poset). Subdivision is central to the theory, and barycentric subdivision of a poset is WHE to the face poset of its order complex. Categories connect up since the second subdivision of a category is a poset, which helps illuminate Thomason's equivalence between the homotopy categories of $\mathcal{C}at$ and $s\mathcal{S}et$.

Weak and actual homotopy equivalences are wildly different for $A$-spaces. In the usual world of spaces, they correspond to homotopy equivalences and simple homotopy equivalences, respectively, a point of view that Barmak's book focuses on. The $n$-sphere is WHE to a space with $2n+2$ points, and that is the minimum number possible.

If the poset $\mathcal{A}_pG$ of non-trivial elementary abelian $p$-subgroups of a finite group $G$ is contractible, then $G$ has a normal $p$-subgroup. A celebrated conjecture of Quillen says in this language that if $\mathcal{A}_pG$ is weakly contractible (WHE to a point), then it is contractible and hence $G$ has a normal $p$-subgroup. There are many interesting contractible finite spaces that are not weakly contractible.

These facts just scratch the surface and were nearly all previously known, but there is much that is new in the book, some of it due to students at Chicago where I have taught this material in our REU off and on since 2003. This is ideal material for the purpose. (Obsolete notes and even current ones can be found on my web page by those sufficiently interested to search: Minian, Barmak's thesis advisor in Buenos Aires, found them there and started off work in Argentina based on them.) I apologize for this extended advertisement, but perhaps Paul's question gives me a reasonable excuse.

Answer 2

Peter May has been working on an entire book (or maybe just a comprehensive set of lecture notes?) addressing your exact question (and much more). The preprint version which he shared with me is called

Peter May, "Finite spaces and larger contexts"

but I can't find an online copy and I'm not sure that I should link to mine without his permission. Perhaps you could email him and ask for one?

So I suppose the short answer to your question is that not only is it possible, but it has been done. One of the morals of Peter's book, as far as I can remember, is that the difference between "homotopy-equivalence" and "weak homotopy equivalence" becomes rather drastic when dealing with finite spaces whereas it is largely a non-issue for CW complexes thanks to Whitehead's theorem.

Finally, I'd like to point out that there is a flavor of Morse theory which works directly on the level of partially ordered sets, and hence would be adaptable to finite spaces (but I don't know if anyone has done this already, it certainly wasn't in Peter's preprint). See for instance Chapter 11 of

Dmitry Kozlov, Combinatorial algebraic topology, Springer (2007).