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?