Acyclic models theorem

Acyclic models theorem

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 mathcal{K} be an arbitrary category, and mathcal{CC} be the category of chain complexes of Abelian groups. Let F,V : mathcal{K} to mathcal{CC} be covariant functors so that F_i = V_i = 0 for i < 0.

Assume now that there are mathcal{M}_k subset mathcal{K} for k ge 0 so that F_k has a basis in mathcal{M}_k , so F is a Free functor. Finally, let V be acyclic, which means that H_{k+1}(V(M)) = 0 for M in mathcal{M}_{k+1} cup mathcal{M}_{k+2}.

Then every natural transformation varphi : H_0(F) to H_0(V) is induced by a natural chain map f : F to V. Additionally, f is unique upto natural homotopy.

References

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