As well known, singular homology sends weak homotpy equivalences into isomorphisms in homology. The lecture notes which I am reading mention that this fact is peculiar of singular homology and does not hold true in general for any homology theory (in the sense of Eilenberg and Steenrod). So I was wondering, which is an example of an homology theory which fails to satisfy this property?
An example of an homology theory which fails to send weak equivalences to isomorphism
algebraic-topologyhomology-cohomology
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The "natural" bit here is a red herring, and seeing that it's not relevant makes the question a lot easier.
Indeed at this level of generality ($\mathcal{L}$ an arbitrary category) we can let functors "be anything" and "natural" sort of loses its point. Indeed, it suffices to take $\mathcal{L}$ to be the discrete category on one object.
At that point, a functor $\mathcal{L}\to \mathsf{Comp}$ is just the choice of a complex, and $\mathcal{L}\to \mathbf{Ab}$ just the choice of an abelian group.
A natural isomorphism (or more generally transformation) is then just an isomorphism (or more generally morphism). Hence at this level of generality, a special case of your question becomes :
If I have an isomorphism between the homologies of two complexes, is this isomorphism induced by some (iso)morphism of complexes ?
The answer to that is clearly no. For instance consider a complex $\mathsf{C}$ with only $\mathbb{Z/2Z}$ in position $0$ and $0$'s elsewhere, and $\mathsf{C'}$with $\mathbb{Z}$ in position $-1$ and $0$, the map $\mathbb{Z}\to \mathbb{Z}$ being multiplication by $2$ . Then the homologies of both complexes are isomorphic (they're $\mathbb{Z/2Z}$ in position $0$, and $0$ elsewhere); but there is no nontrivial morphism $\mathsf{C\to C'}$, hence no morphism that could induce the isomorphism in homology.
In fact this isn't category-dependent (you might argue that I chose a dummy category): for any category $\mathcal{L}$, you can "model" this situation by picking constant functors, and the result will be the same; therefore if you want to add some conditions to change the answer to "yes", then the conditions will not only be on $\mathcal{L}$ but also on the functors you have, e.g. respect products or whatever. But that would be another question, and right now I don't have examples of natural (non trivial) constraints one could add on the functors to have a positive answer.
The moral here is that the functor $H_* : \mathsf{Comp}\to \mathbf{Ab}$ is very very far from being full.
No.
The "classical" Eilenberg-Steenrod axioms are homotopy invariance, exactness, excision and dimension. These describe ordinary homology theories. It is well-known that for finite CW-pairs the homology groups $H_n(X,A)$ are (up to natural isomorphism) uniquely determined by the coefficient group $G = H_0(*)$, where $*$ is a one-point space. In fact, they agree with the singular homology groups of $(X,A)$ with coefficients in $G$. In particular $H_n(X,A) = 0$ for $n < 0$.
Beyond finite CW-pairs things are more sophisticated. In
I.M. James and J.H.C. Whitehead, "Homology with zero coefficients", The Quarterly Journal of Mathematics, Volume 9, Issue 1, 1958, Pages 317–320
one can find examples of non-trivial ordinary homology theories with zero coefficient group. There are infinite CW-complexes $X$ with nonvanishing homology groups. Making a dimension shift ($H'_n = H_{n+k}$ for some $k \in \mathbb N$) we can achieve that $H'_n(X) \ne 0$ for negative $n$. The theory $H'_*$ has coefficient group zero, if you do not like that consider $H'_* \oplus H^{sing}_*$, where $H^{sing}_*$ is singular homology with $\mathbb Z$-coefficients.
Best Answer
If one is explicitly mentioning E.S. axioms, I would find it hard to argue that you can interpret the question so there are any examples. When one mentions the E.S. axioms one almost always either restricts to CW complexes, in which case weak homotopy equivalences are equivalent to homotopy equivalences, or one explicitly requires that weak homotopy equivalences are sent to isomorphisms.
Now there are certainly collections of groups that you can assign to a topological space that many would be happy to call a homology theory, but they do not satisfy all the E.S. axioms. In these cases, they might not send weak equivalences to isomorphisms. Switching to cohomology for the sake of an example: Cech cohomology does not send weak equivalences to equivalences, for example, the topologist's sine curve has nontrivial first cohomology when it has the weak homotopy type of two points.