nLab category of chain complexes

Definition

Let $\mathcal{A}$ be an additive category.

Recall the notion of chain complex, of chain map between chain complexes and of chain homotopy between chain maps in $\mathcal{A}$.

Call a chain complex $C_\bullet$

• bounded below if there is $k \in \mathbb{N}$ such that $C_{n \leq k} = 0$;

• bounded above if there is $k \in \mathbb{N}$ such that $C_{n \geq k} = 0$;

• bounded if it is bounded below and bounded above. We have

Definition

Write $Ch_\bullet(\mathcal{A})$ for the category whose objects are chain complexes in $\mathcal{A}$ and whose morphisms are chain maps between these.

This is the category of chain complexes in $\mathcal{A}$.

Several variants of this category are of relevance.

Definition

Write $Ch_\bullet^{+,-,b}(\mathcal{A}) \hookrightarrow Ch_\bullet(\mathcal{A})$ for the full subcategory on the chain complexes which are, respectively, bounded above, bounded below or bounded.

Definition

Write $K(\mathcal{A})$ for the category obtained from $Ch_\bullet(\mathcal{A})$ by identifying homotopic chain maps.

$$K(\mathcal{A})(C_\bullet, D_\bullet) \coloneqq Ch_\bullet(C_\bullet, D_\bullet)/chain-homotopy \,.$$

Accordingly $K^{+,-,b}(\mathcal{A}) \hookrightarrow K(\mathcal{A})$ denotes the full subcategory on the chain complexes bounded above, bounded below or bounded, respectively.

This is sometimes called the homotopy category of chain complexes. But see the warning on terminology there, as this term is also appropriate for the category in the following remark.

Remark

If $\mathcal{A}$ is moreover an abelian category, then there is also the derived category $D(\mathcal{A})$, obtained from $Ch_\bullet(\mathcal{A})$ or $K(\mathcal{A})$ by universally inverting all quasi-isomorphisms. See at derived category for more on this.

Properties

Abelian structure

Theorem

For $\mathcal{A}$ an abelian category also the category of chain complexes $Ch_\bullet(\mathcal{A})$ is again an abelian category.

We discuss the ingredients that go into this statement.

(…)

Proposition

For $f : C_\bullet \to D_\bullet$ a chain map,

• the complex $ker(f)$ of degreewise kernels in $\mathcal{A}$ is the kernel of $f$ in $Ch_\bullet(\mathcal{A})$;

• the complex $coker(f)$ of degreewise cokernels in $\mathcal{A}$ is the cokernel of $f$ in $Ch_\bullet(\mathcal{A})$.

Remark

A sequence of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$ is a short exact sequence in $Ch_\bullet(\mathcal{A})$ precisely if each component $0 \to A_n \to B_n \to C_n \to 0$ is a short exact sequence in $\mathcal{A}$.

(…)

In fact:

Proposition

When $\mathcal{A}$ is a Grothendieck abelian category then so is $Ch_\bullet(\mathcal{A})$.

(e.g. Hovey (1999), p. 3, see also this example at Grothendieck category).

Since every Grothendieck abelian category is locally presentable (Beke 2000, Prop. 3.10, see this example), it follows that:

Corollary

When $\mathcal{A}$ is a Grothendieck abelian category then its category of chain complexes $Ch_\bullet(\mathcal{A})$ is locally presentable.

Example

The assumption in Prop. is fulfilled in the usual situation of chain complexes of $R$-modules (e.g.: of vector spaces, when $R$ is a field), since for any commutative ring $R$ the category $R$Mod is a Grothendieck abelian category (by this example).

Closed monoidal structure

Proposition

Equipped with the standard tensor product of chain complexes $\otimes$ the category of chain complexes is a monoidal category $(Ch_\bullet(R Mod), \otimes)$. The unit object is the chain complex concentrated in degree 0 on the tensor unit $R$ of $R Mod$.

Proposition

In fact $(Ch_\bullet(R Mod), \otimes)$ is a closed monoidal category, the internal hom is the standard internal hom of chain complexes.

Related concepts

• homotopy category of chain complexes

• model structure on chain complexes

• derived category, derived functor

• (∞,1)-category of chain complexes

• equivariant chain complex

References

The closed monoidal category-structure on chain complexes:

• Samuel Eilenberg, G. Max Kelly, §IV.6 of: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]

Textbook account:

• Charles Weibel, Chapter 1 in: An Introduction to Homological Algebra, Cambridge University Press (1994) [doi:10.1017/CBO9781139644136, pdf]

Discussion in the context of model structures on chain complexes:

• Mark Hovey, p. 3 of: Model category structures on chain complexes of sheaves (1999) [K-theory:0366, pdf, pdf]

category: category