From a course in commutative algebra I know the algebraic way to define a homology theory by taking pointwise quotients of chain complexes (which can for example be obtained by taking projective resolutions of some sort).
In a course on algebraic topology, which I am currently attending, we introduced generalized relative extraordinary homology theories as sequence of functors $H_n:\operatorname{hTop}^\hookrightarrow \rightarrow \operatorname{Mod}_R^{\mathbb{Z}gr}$ together with natural transformations $\partial_n:H_n \implies H_{n-1}\circ R$ satisfying the Eilenberg-Steenrod axioms (here $\operatorname{hTop}^\hookrightarrow$ denotes the category of pairs of subspaces $X \supseteq A$ and morphisms $f:(X,A) \rightarrow (Y,B)$ which satisfy $f(A) \subseteq B$; $R$ is the functor given by $(X,A) \mapsto (X,\emptyset)$). The corresponding reduced homology theory is then given by $\tilde{H}(X) = \ker (H(X,\emptyset) \rightarrow H(*,\emptyset))$.
My question now is:
Is there a unified treatment of both the algebraic and the topological notions of homology theory (relative or reduced)?
Initially I thought generalized homology theories could be obtained from homologies of certain chain complexes, as for example the simplicial homology is. However a result from Bauer seems to show that this is not true in general and those theories, who come from chain complexes, are closely related to ordinary homology theories. Extraordinary homology theories seem to be very important however (Wikipedia mentions important sounding words like topological K-theory and cobordism), so it seems to me that this is the wrong approach to making the notions agree...
Yet, the Eilenberg-Steenrod axioms don't seem to rely on having topological spaces as underlying category. I wonder, wether the algebraic homology functors can be expressed in a similar set of axioms, leaving chain complexes and derived functors just as a tool to obtain homology theories...
Thank you very much for your time and patience!
This is more of a long comment addressing both (topological) homology theories coming from chain complexes and homology theories for chain complexes:
Let's suppose that your homology theory is defined for CW pairs. One way to attempt to find a chain complex whose homology is the homology of your space is to take the cellular filtration of your CW complex, and two each pair $(X_n,X_{n-1})$ we may form the long exact sequence for the homology theory which we can piece together to form a spectral sequence called the Atiyah-Hirzebruch spectral sequence. This sequence starts with the relative homology of your complex with coefficients in the extraordinary homology of a point and converges to the extraordinary homology of your complex. The obstruction to this spectral sequence simply being a chain complex is if the extraordinary homology of a point has multiple dimensions nontrivial.
That is not to say that the homology theory can't arise from a chain complex if the homology of a point is nontrivial in more than one dimension. For example, unoriented bordism is maps from smooth, compact n-manifolds into your space modulo the relation of bordism which says two maps are equivalent if there is a larger map on a manifold which restricts to the disjoint union of the maps on the boundary. This can be written in terms of a chain complex (of monoids) because the boundary of a manifold has no boundary. Very surprisingly, it turns out that this homology theory is actually a direct sum of shifted $\mathbb{Z}/2$-homology, but for other bordism theories this is not the case.
I'm sure much is known about what types of homology theories factor through chain complexes, I would not be surprised, for example, if any theory which outputted rational vector spaces factored through a map to rational chain complexes.
For your question about homology theories on chain complexes, I believe these all come from tensoring with a chain complex and then taking its homology. This should follow from the fact that the stabilization of the category of chain complexes is itself. Here this just means that shift up and shift down are already inverses, and these play the role of suspension and inverse suspension in stable homotopy theory. For more information, look up hypercohomology.