nLab
derived category

Idea

For $C$ an abelian category, notice that naturally associated to $C$ is

• the category of chain complexes $K(C)$ in $C$ which is naturally a homotopical category;

• the stable ∞-category $K_{\mathrm{\infty }}(C)$ of chain complexes in $C$.

The derived category $D(C)$ of $C$ is equivalently

• the 1-categorical homotopy category of $K(C)$;

• the (∞,1)-categorical homotopy category of $K_{\mathrm{\infty }}(C)$.

In either case, this means that under the canonical localization functor

$$Q:K(C)\to D(C)$$Q : K(C) \to D(C)

the quasi-isomorphisms of chain complexes become true isomorphisms and that $D(C)$ is universal with respect to this property.

Definition

Let $C$ be an abelian category and $K(C)$ its category of chain complexes modulo chain homotopy?.

Equip $K(C)$ with the structure of a homotopical category by declaring the weak equivalences to be the quasi-isomorphisms: those morphisms $f:V\to W$ which induce isomorphisms in homology, $H(f):H(V)\stackrel{\simeq }{\to }H(W)$.

The derived category $D(C)$ is the homotopy category of $K(C)$ with respect to these weak equivalences.

Remark

This is a special case of the construction of a homotopy category of a triangulated category with respect to a null system.

Let $N(C)\subset K(C)$ be the full subcategory of $K(C)$ on those chain complexes $V$ whose homology vanishes, $H(V)=0$. Then $f:V\to W$ is a quasi-isomorphism iff there exists a distinguished triangle

$$V\stackrel{f}{\to }W\to \mathrm{cone}(f)$$V \stackrel{f}{\to} W \to cone(f)

with the mapping cone $\mathrm{cone}(f)\in N(C)$.

The derived category is still naturally a triangulated category itself.

References

A disucssion in a comprehensive category theoretic and homological algebra-context is in section 13nd Sheaves) of

• Kashiwara-Schapira, Categories and Sheaves

A pedagogical introduction is

• R. P. Thomas, Derived categories for the working mathematician (arXiv)

A good survey of the more general topic of derived categories is

• Bernhard Keller, Derived categories and their uses (ps)

See in particular the list of references given there.

For a discussion in the context of (∞,1)-categories and in particular stable (∞,1)-categories see section 13, p. 53

of

• Jacob Lurie, Stable ∞-Categories