Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, indexed by $\Bbb Z$) from spaces into abelian groups as the homology of a funcotrial chain complex: just take the graded abelian group that is the sum of the groups with zero differential. As you'd expect, I don't find that satisfying. I could (possibly) rule such silliness out by adding some freeness hypothesis on the chain complex functors, but I don't necessarily wish to do that.
Similarly, is there any a priori reason one may expect to be able to calculate the homology of a loop space $\Omega X$ given the (singular or cellular) chain complex of the space $X$ (with some connectedness hypothesis)? In a way this is a silly question : as I understand it, one can completely reconstruct the original space $X$ (up to weak homotopy equivalence) from its singular simplicial set $(\mathrm{map}(\Delta^n,X))_{n\in\Bbb N}$, so surely its singular chain complex carries much more information than is expressed in its homology. Still I'm baffled that an algebraic procedure, the cobar construction, can under some mild hypothesis of connectedness, produce a chain complex model for the homology of the loop space, which is a topological construction.
Maybe I should phrase my question thusly : why is it that there are good functorial chain complexes that allow one to calculate various groups associated to a space and perform topological constructions algebraically, and should we expect it?
Later addition. The original motivation behind this question comes from rational homotopy theory which I'm only beginning to learn. The rational homotopy groups of a space suitably shifted in index form a (sign-)graded Lie algebra. They are all expressible as the homology of a differential graded Lie algebra. I hope this example makes my question clearer : I never expected the rational homotopy groups of a space to be homology groups of chain complexes in an interesting way.