In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic.
The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane.
[1] They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes.
Eilenberg and MacLane then discovered the theorem to generalize this process.
It can be used to prove the Eilenberg–Zilber theorem; this leads to the idea of the model category.
be covariant functors such that: Then the following assertions hold:[2][3] What is above is one of the earliest versions of the theorem.
is a complex of projectives in an abelian category and
is an acyclic complex in that category, then any map
This specializes almost to the above theorem if one uses the functor category
Free functors are projective objects in that category.
The morphisms in the functor category are natural transformations, so the constructed chain maps and homotopies are all natural.
is basically just a free (and hence projective) module.
being acyclic at the models (there is only one) means nothing else than that the complex
be an abelian category (for example,
will be called an acyclic class provided that: There are three natural examples of acyclic classes, although doubtless others exist.
In functor categories (e.g. the category of all functors from topological spaces to abelian groups), there is a class of complexes that are contractible on each object, but where the contractions might not be given by natural transformations.
Another example is again in functor categories but this time the complexes are acyclic only at certain objects.
denote the class of chain maps between complexes whose mapping cone belongs to
does not necessarily have a calculus of either right or left fractions, it has weaker properties of having homotopy classes of both left and right fractions that permit forming the class
The boundary operator is given by We say that the chain complex functor
the corresponding class of arrows in the category of chain complexes.
to a natural transformation of chain functors
Here is an example of this last theorem in action.
be the category of triangulable spaces and
be the category of abelian group valued functors on
be the singular chain complex functor and
be the simplicial chain complex functor.
is presentable and acyclic is not entirely straightforward and uses a detour through simplicial subdivision, which can also be handled using the above theorem).
and so we conclude that singular and simplicial homology are isomorphic on
There are many other examples in both algebra and topology, some of which are described in [4][5]