nLab category of chain complexes



Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




Let π’œ\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 π’œ\mathcal{A}.

Call a chain complex C β€’C_\bullet

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

  • bounded above if there is kβˆˆβ„•k \in \mathbb{N} such that C nβ‰₯k=0C_{n \geq k} = 0;

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


Write Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) for the category whose objects are chain complexes in π’œ\mathcal{A} and whose morphisms are chain maps between these.

This is the category of chain complexes in π’œ\mathcal{A}.

Several variants of this category are of relevance.


Write Ch β€’ +,βˆ’,b(π’œ)β†ͺCh β€’(π’œ)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.


Write K(π’œ)K(\mathcal{A}) for the category obtained from Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) by identifying homotopic chain maps.

K(π’œ)(C β€’,D β€’)≔Ch β€’(C β€’,D β€’)/chainβˆ’homotopy. K(\mathcal{A})(C_\bullet, D_\bullet) \coloneqq Ch_\bullet(C_\bullet, D_\bullet)/chain-homotopy \,.

Accordingly K +,βˆ’,b(π’œ)β†ͺK(π’œ)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.


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


Abelian structure


For π’œ\mathcal{A} an abelian category also the category of chain complexes Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) is again an abelian category.

We discuss the ingredients that go into this statement.



For f:C β€’β†’D β€’f : C_\bullet \to D_\bullet a chain map,

  • the complex ker(f)ker(f) of degreewise kernels in π’œ\mathcal{A} is the kernel of ff in Ch β€’(π’œ)Ch_\bullet(\mathcal{A});

  • the complex coker(f)coker(f) of degreewise cokernels in π’œ\mathcal{A} is the cokernel of ff in Ch β€’(π’œ)Ch_\bullet(\mathcal{A}).


A sequence of chain complexes 0β†’A β€’β†’B β€’β†’C β€’β†’00 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 is a short exact sequence in Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) precisely if each component 0β†’A nβ†’B nβ†’C nβ†’00 \to A_n \to B_n \to C_n \to 0 is a short exact sequence in π’œ\mathcal{A}.


In fact:


When π’œ\mathcal{A} is a Grothendieck abelian category then so is Ch β€’(π’œ)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:


When π’œ\mathcal{A} is a Grothendieck abelian category then its category of chain complexes Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) is locally presentable.


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

Closed monoidal structure


Equipped with the standard tensor product of chain complexes βŠ—\otimes the category of chain complexes is a monoidal category (Ch β€’(RMod),βŠ—)(Ch_\bullet(R Mod), \otimes). The unit object is the chain complex concentrated in degree 0 on the tensor unit RR of RModR Mod.


In fact (Ch β€’(RMod),βŠ—)(Ch_\bullet(R Mod), \otimes) is a closed monoidal category, the internal hom is the standard internal hom of chain complexes.


The closed monoidal category-structure on chain complexes:

Textbook account:

Discussion in the context of model structures on chain complexes:

category: category

Last revised on December 16, 2023 at 17:45:52. See the history of this page for a list of all contributions to it.