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".
If your homology theory is of the form $H_n(X) = H_n(S(X))$ 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))$\Gamma(S(X))$ and hence (by realization) a topological space in which you are "counting holes".
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".