In Algebraic Topology
, the Method of Acyclic Models, or Acyclic Models Theorem describes a process by which two homology theories
can be shown to be 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.
Statement of the Theorem
Let be an arbitrary category, and be the category of chain complexes of Abelian groups. Let be covariant functors so that for .
Assume now that there are for so that has a basis in , so is a Free functor. Finally, let be acyclic, which means that for .
Then every natural transformation is induced by a natural chain map . Additionally, is unique upto natural homotopy.