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 . Let be an arbitrary and be the category of chain complexes of -modules over some ring . Let be covariant functors such that: for . There are for such that has a basis in , so is a free functor. is - and -acyclic at these models, which means that for all and all . Then the following assertions hold: Every natural transformation induces a natural chain map . If are natural transformations, are natural chain maps as before and for all models , then there is a natural chain homotopy between and . In particular the chain map is unique up to natural chain homotopy. What is above is one of the earliest versions of the theorem. Another version is the one that says that if is a complex of projectives in an and is an acyclic complex in that category, then any map extends to a chain map , 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, 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 a category with only one object. Then the free functor 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 is acyclic.