Projective and acyclic complexes

What is above is one of the earliest versions of the theorem. Another version is the one that says that if ${\displaystyle K}$ is a complex of projectives in an abelian category and ${\displaystyle L}$ is an acyclic complex in that category, then any map ${\displaystyle K_{0}\to L_{0}}$ extends to a chain map ${\displaystyle K\to L}$, unique up to homotopy.

This specializes almost to the above theorem if one uses the functor category ${\displaystyle {\mathcal {C}}(R)^{\mathcal {K}}}$ 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, ${\displaystyle V}$ 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 ${\displaystyle {\mathcal {K}}}$ a category with only one object. Then the free functor ${\displaystyle F}$ is basically just a free (and hence projective) module. ${\displaystyle V}$ being acyclic at the models (there is only one) means nothing else than that the complex ${\displaystyle V}$ is acyclic.

Acyclic classes

There is a grand theorem that unifies both of the above.^[4][5] Let ${\displaystyle {\mathcal {A}}}$ be an abelian category (for example, ${\displaystyle {\mathcal {C}}(R)}$ or ${\displaystyle {\mathcal {C}}(R)^{\mathcal {K}}}$). A class ${\displaystyle \Gamma }$ of chain complexes over ${\displaystyle {\mathcal {A}}}$ will be called an acyclic class provided that:

• The 0 complex is in ${\displaystyle \Gamma }$.
• The complex ${\displaystyle C}$ belongs to ${\displaystyle \Gamma }$ if and only if the suspension of ${\displaystyle C}$ does.
• If the complexes ${\displaystyle K}$ and ${\displaystyle L}$ are homotopic and ${\displaystyle K\in \Gamma }$, then ${\displaystyle L\in \Gamma }$.
• Every complex in ${\displaystyle \Gamma }$ is acyclic.
• If ${\displaystyle D}$ is a double complex, all of whose rows are in ${\displaystyle \Gamma }$, then the total complex of ${\displaystyle D}$ belongs to ${\displaystyle \Gamma }$.

There are three natural examples of acyclic classes, although doubtless others exist. The first is that of homotopy contractible complexes. The second is that of acyclic complexes. In functor categories (e.g. the category of all functors from topological spaces to abelian groups), there is a class of complexes that are contractible on each object, but where the contractions might not be given by natural transformations. Another example is again in functor categories but this time the complexes are acyclic only at certain objects.

Let ${\displaystyle \Sigma }$ denote the class of chain maps between complexes whose mapping cone belongs to ${\displaystyle \Gamma }$. Although ${\displaystyle \Sigma }$ does not necessarily have a calculus of either right or left fractions, it has weaker properties of having homotopy classes of both left and right fractions that permit forming the class ${\displaystyle \Sigma ^{-1}C}$ gotten by inverting the arrows in ${\displaystyle \Sigma }$.^[4]

Let ${\displaystyle G}$ be an augmented endofunctor on ${\displaystyle C}$, meaning there is given a natural transformation ${\displaystyle \epsilon :G\to Id}$ (the identity functor on ${\displaystyle C}$). We say that the chain complex ${\displaystyle K}$ is ${\displaystyle G}$-presentable if for each ${\displaystyle n}$, the chain complex

${\displaystyle \cdots K_{n}G^{m+1}\to K_{n}G^{m}\to \cdots \to K_{n}}$

belongs to ${\displaystyle \Gamma }$. The boundary operator is given by

${\displaystyle \sum (-1)^{i}K_{n}G^{i}\epsilon G^{m-i}:K_{n}G^{m+1}\to K_{n}G^{m}}$.

We say that the chain complex functor ${\displaystyle L}$ is ${\displaystyle G}$-acyclic if the augmented chain complex ${\displaystyle L\to H_{0}(L)\to 0}$ belongs to ${\displaystyle \Gamma }$.

Theorem. Let ${\displaystyle \Gamma }$ be an acyclic class and ${\displaystyle \Sigma }$ the corresponding class of arrows in the category of chain complexes. Suppose that ${\displaystyle K}$ is ${\displaystyle G}$-presentable and ${\displaystyle L}$ is ${\displaystyle G}$-acyclic. Then any natural transformation ${\displaystyle f_{0}:H_{0}(K)\to H_{0}(L)}$ extends, in the category ${\displaystyle \Sigma ^{-1}(C)}$ to a natural transformation of chain functors ${\displaystyle f:K\to L}$ and this is unique in ${\displaystyle \Sigma ^{-1}(C)}$ up to chain homotopies. If we suppose, in addition, that ${\displaystyle L}$ is ${\displaystyle G}$-presentable, that ${\displaystyle K}$ is ${\displaystyle G}$-acyclic, and that ${\displaystyle f_{0}}$ is an isomorphism, then ${\displaystyle f}$ is homotopy equivalence.