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. 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 K {displaystyle {mathcal {K}}} be an arbitrary category and C ( R ) {displaystyle {mathcal {C}}(R)} be the category of chain complexes of R {displaystyle R} -modules. Let F , V : K → C ( R ) {displaystyle F,V:{mathcal {K}} o {mathcal {C}}(R)} be covariant functors such that: Then the following assertions hold: What is above is one of the earliest versions of the theorem. Anotherversion is the one that says that if K {displaystyle K} is a complex ofprojectives in an abelian category and L {displaystyle L} is an acycliccomplex in that category, then any map K 0 → L 0 {displaystyle K_{0} o L_{0}} extends to a chain map K → L {displaystyle K o L} , unique up tohomotopy. This specializes almost to the above theorem if one uses the functor category C ( R ) K {displaystyle {mathcal {C}}(R)^{mathcal {K}}} 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 {displaystyle 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 K {displaystyle {mathcal {K}}} a category with only one object. Then the free functor F {displaystyle F} is basically just a free (and hence projective) module. V {displaystyle V} being acyclic at the models (there is only one) means nothing else than that the complex V {displaystyle V} is acyclic. There is a grand theorem that unifies both of the above. Let A {displaystyle {mathcal {A}}} be an abelian category (for example, C ( R ) {displaystyle {mathcal {C}}(R)} or C ( R ) K {displaystyle {mathcal {C}}(R)^{mathcal {K}}} ). A class Γ {displaystyle Gamma } of chain complexes over A {displaystyle {mathcal {A}}} will be called an acyclic class provided that: