My question concerns finite sets carrying a not-necessarily-discrete topology. I'm wondering if there's an analogue for homotopy theory where the role of $S^n$ is played by some other, finite set. (My preliminary thoughts involved cyclic groups with the discrete topology playing the role of $S^1$, but I can't be sure yet if this goes anywhere.)
Reading up, I found that, perhaps unsurprisingly, discrete homotopy theories for graphs have been studied, for example in G. Malle, A Homotopy Theory for Graphs, Glasnik Matematicki, 18 (1983), 3-25, and they exhibit precisely the behavior one expects of them. But, this is something of a proper subset of what I'm looking for, and in many ways the theories are of a "less topological" flavor than I would prefer.
This is primarily a reference request, but if anyone has ideas or examples I should be keeping in mind I would be equally appreciative for those.
$S^1$ is actually weakly homotopy equivalent to the pseudocircle $P$. It is therefore possible to replace $S^1$ with a finite space in many cases, for example the fundamental group $[S^1,X]$ is naturally isomorphic to $[P,X]$ (for all, not just finite $X$). A similar construction works for the higher homotopy groups as any finite simplicial complex (like the $n$-sphere) is weakly homotopy equivalent to a finite topological space.