Acyclic model

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. 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.
Statement of the theorem.
Let formula_1 be an arbitrary category and formula_2 be the category of chain complexes of formula_3-modules over some ring formula_3. Let formula_5 be covariant functors such that:
Then the following assertions hold:
Generalizations.
Projective and acyclic complexes.
What is above is one of the earliest versions of the theorem. Another
version is the one that says that if formula_28 is a complex of
projectives in an abelian category and formula_29 is an acyclic
complex in that category, then any map formula_30 extends to a chain map formula_31, unique up to
homotopy.
This specializes almost to the above theorem if one uses the functor category formula_32 as the abelian 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. The difference is that in the above version, formula_13 being acyclic is a stronger assumption than being acyclic only at certain objects.
On the other hand, the above version almost implies this version by letting formula_1 a category with only one object. Then the free functor formula_12 is basically just a free (and hence projective) module. formula_13 being acyclic at the models (there is only one) means nothing else than that the complex formula_13 is acyclic.
Acyclic classes.
There is a grand theorem that unifies both of the above. Let formula_38 be an abelian category (for example, formula_2 or formula_32). A class formula_41 of chain complexes over formula_38 will be called an acyclic class provided that:
There are three natural examples of acyclic classes, although doubtless others exist. The first is that of homotopy contractible complexes. The second is that of acyclic complexes. 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.
Let formula_56 denote the class of chain maps between complexes whose mapping cone belongs to formula_41. Although formula_56 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 formula_59 gotten by inverting the arrows in formula_56.
Let formula_61 be an augmented endofunctor on formula_44, meaning there is given a natural transformation
formula_63 (the identity functor on formula_44). We say that the chain complex formula_28 is formula_61-"presentable" if for each formula_67, the chain complex
belongs to formula_41. The boundary operator is given by
We say that the chain complex functor formula_29 is formula_61-"acyclic" if the augmented chain complex
formula_73 belongs to formula_41.
Theorem. "Let formula_41 be an acyclic class and formula_56 the corresponding class of arrows in the category of chain complexes. Suppose that formula_28 is formula_61-presentable and formula_29 is formula_61-acyclic. Then any natural transformation formula_81 extends, in the category formula_82 to a natural transformation of chain functors formula_83 and this is
unique in formula_82 up to chain homotopies. If we suppose, in addition, that formula_29 is formula_61-presentable, that formula_28 is formula_61-acyclic, and that formula_89 is an isomorphism, then formula_25 is homotopy equivalence.
Example.
Here is an example of this last theorem in action. Let formula_91 be the category of triangulable spaces and formula_44 be the category of abelian group valued functors on formula_91. Let
formula_28 be the singular chain complex functor and formula_29 be the simplicial chain complex functor. Let formula_96 be the functor that assigns to each space formula_91 the space
Here, formula_99 is the formula_67-simplex and this functor assigns to formula_91 the sum of as many copies of each formula_67-simplex as there are maps formula_103. Then let formula_61 be defined by formula_105. There is an obvious augmentation formula_106 and this induces one on formula_61. It can be shown that both formula_28 and formula_29 are both formula_61-presentable and formula_61-acyclic (the proof that formula_29 is presentable and acyclic is not entirely straightforward and uses a detour through simplicial subdivision, which can also be handled using the above theorem). The class formula_41 is the class of homology equivalences. It is rather obvious that formula_114 and so we conclude that singular and simplicial homology are isomorphic on formula_91.
There are many other examples in both algebra and topology, some of which are described in 