[Math] Other Homology Theories still Count Holes

Question

This may be a naive question, but since first learning homology I considered it as a tool which counts appropriate holes in your space (on top of orientation and torsion phenomena). Then I was introduced to homology of groups and now Floer/Morse homologies. Do these homologies still count "holes" in some fashion?
In the case of group homology, $H_\ast(G)\cong H_\ast(BG)$, so we can view this homology as a count of holes in the Milnor construction (CW-complex assembled from points in the discrete group with the group structure).
In Floer homology we're counting holomorphic curves (flow-lines in Morse homology), but it isn't viewed as having these curves "wrap around holes", so I am not sure if this hole-detecting view of homology breaks down.

[[Edit]]: I will narrow down my question. Are there instances where I can treat $HF_\ast$ as $H_\ast$ of a particular space? For instance, I just realized that with nice conditions we have $HF^\ast(L,L)=H^\ast(L)$ in Lagrangian-Floer homology, so here it counts the holes of the Lagrangian submanifold.
Thanks to Steven Landsburg's response, we can usually find such a space (but ideally would be looking for something explicit, such as Floer homotopy type with $SH_\ast(T^\ast M)=H_\ast(\mathcal{L}M)$).

Answer 1

If your homology theory is of the form $H_n(X) = H_n(S(X)) $ where $S$ is some functor from your original category to non-negative chain complexes, then the Dold-Kan correspondence gives you a corresponding simplicial abelian group $\Gamma(S(X)) $ and hence (by realization) a topological space in which you are "counting holes".