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.

Let ${\displaystyle {\mathcal {K}}}$ be an arbitrary category and ${\displaystyle {\mathcal {C}}(R)}$ be the category of chain complexes of ${\displaystyle R}$-modules. Let ${\displaystyle F,V:{\mathcal {K}}\to {\mathcal {C}}(R)}$ be covariant functors such that:

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