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 \mathcal{K} be an arbitrary category and be the category of chain complexes of R-modules over some ring R. Let 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 K is a complex of projectives in an abelian category and L is an acyclic complex in that category, then any map extends to a chain map K\to L, unique up to homotopy. This specializes almost to the above theorem if one uses the functor category 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, V 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 \mathcal{K} a category with only one object. Then the free functor F is basically just a free (and hence projective) module. V being acyclic at the models (there is only one) means nothing else than that the complex V is acyclic.

Acyclic classes

There is a grand theorem that unifies both of the above. Let \mathcal{A} be an abelian category (for example, or ). A class \Gamma of chain complexes over \mathcal{A} 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 \Sigma denote the class of chain maps between complexes whose mapping cone belongs to \Gamma. Although \Sigma 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 gotten by inverting the arrows in \Sigma. Let G be an augmented endofunctor on C, meaning there is given a natural transformation (the identity functor on C). We say that the chain complex K is G-presentable if for each n, the chain complex belongs to \Gamma. The boundary operator is given by We say that the chain complex functor L is G-acyclic if the augmented chain complex belongs to \Gamma. Theorem. Let \Gamma be an acyclic class and \Sigma the corresponding class of arrows in the category of chain complexes. Suppose that K is G-presentable and L is G-acyclic. Then any natural transformation extends, in the category to a natural transformation of chain functors f:K\to L and this is unique in up to chain homotopies. If we suppose, in addition, that L is G-presentable, that K is G-acyclic, and that f_0 is an isomorphism, then f is homotopy equivalence.

Example

Here is an example of this last theorem in action. Let X be the category of triangulable spaces and C be the category of abelian group valued functors on X. Let K be the singular chain complex functor and L be the simplicial chain complex functor. Let E: X\to X be the functor that assigns to each space X the space Here, \Delta_n is the n-simplex and this functor assigns to X the sum of as many copies of each n-simplex as there are maps. Then let G be defined by G(C)=CE. There is an obvious augmentation EX\to X and this induces one on G. It can be shown that both K and L are both G-presentable and G-acyclic (the proof that L 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 \Gamma is the class of homology equivalences. It is rather obvious that and so we conclude that singular and simplicial homology are isomorphic on X. There are many other examples in both algebra and topology, some of which are described in