nLab chain complex

Idea

A chain complex is a complex in an additive category (often assumed to be an abelian category).

The archetypical example, from which the name derives, is the singular chain complex $C_\bullet(X)$ of a topological space $X$.

Chain complexes are the basic objects of study in homological algebra.

Basic

A chain complex $V_\bullet$ is a sequence $\{V_n\}_{n \in \mathbb{Z}}$ of abelian groups or modules (for instance vector spaces) or similar equipped with linear maps $\{d_n : V_{n+1} \to V_n\}$ such that $d^2 = 0$, i.e. the composite of two consecutive such maps is the zero morphism $d_n \circ d_{n+1} = 0$.

A basic example is the singular chain complex of a topological space, or the de Rham complex of a smooth manifold. Another type of example occurs with the Dold-Kan correspondence as the Moore complex of a simplicial abelian group or similar. Both the first and the third of these types of example correspond, on the surface, to chain complexes in which the grading is by $\mathbb{N}$, not $\mathbb{Z}$. Dually the de Rham complex example can be included by indexing by the non-positive integers, but by defining them to take trivial, that is zero, values in other dimensions they become chain complexes in the sense used here. The more general definition is important as it is (i) more inclusive and (ii) leads to objects that behave well with respect to shift / translation operators, (see below).

Chain complexes crucially come with their chain homology groups. When regarding chain maps between them that induce isomorphisms on these groups (quasi-isomorphisms) as weak equivalences, chain complexes form a useful presentation for aspects of stable homotopy theory. More on this aspect below.

Meaning in homotopy theory

By the Dold-Kan correspondence there is an equivalence between the category of connective chain complexes of abelian groups and the category of abelian simplicial groups. The functor

$$NCC:AB^\Delta^{op}\to Ch_\bullet^+(AB)$$

giving this equivalence is called normalized chain complex functor or Moore complex functor.

In particular this reasoning shows that connective chain complexes of abelian groups are a model for abelian ∞-groups (aka Kan complexes with abelian group structure). This correspondence figures as part of the cosmic cube-classification scheme of n-categories.

Definition

In components

Let $\mathcal{C}$ be an abelian category.

Definition

A ($\mathbb{Z}$-graded) chain complex in $\mathcal{C}$ is

• a collection of objects $\{C_n\}_{n\in\mathbb{Z}}$,

• and of morphisms $\partial_n : C_n \to C_{n-1}$

$$\cdots \overset{\partial_3}{\to} C_2 \overset{\partial_2}{\to} C_1 \overset{\partial_1}{\to} C_0 \overset{\partial_0}{\to} C_{-1} \overset{\partial_{-1}}{\to} \cdots$$

such that

$$\partial_n \circ \partial_{n+1} = 0$$

(the zero morphism) for all $n \in \mathbb{Z}$.

A homomorphism of chain complexes is a chain map (see there). Chain complexes with chain maps between them form the category of chain complexes $Ch_\bullet(\mathcal{C})$.

One uses the following terminology for the components of a chain complex, all deriving from the example of a singular chain complex:

Definition

For $C_\bullet$ a chain complex

• the morphisms $\partial_n$ are called the differentials or boundary maps;

• the elements of $C_n$ are called the $n$-chains;

• the elements in the kernel

  $$Z_n \coloneqq ker(\partial_{n-1})$$

  of $\partial_{n-1} : C_n \to C_{n-1}$ are called the $n$-cycles;

• the elements in the image

  $$B_n \coloneqq im(\partial_n)$$

  of $\partial_{n} : C_{n+1} \to C_{n}$ are called the $n$-boundaries;

Notice that due to $\partial \partial = 0$ we have canonical inclusions

$$0 \hookrightarrow B_n \hookrightarrow Z_n \hookrightarrow C_n \,.$$

• the cokernel

  $$H_n \coloneqq Z_n/B_n$$

  is called the degree-$n$ chain homology of $C_\bullet$.

$$0 \to B_n \to Z_n \to H_n \to 0 \,.$$

The dual notion:

Definition

A cochain complex in $\mathcal{C}$ is a chain complex in the opposite category $\mathcal{C}^{op}$. Hence a tower of objects and morphisms as above, but with each differential $d_n : V^n \to V^{n+1}$ increasing the degree.

Remark

One also says homologically graded complex, for the case that the differentials lower degree, and cohomologically graded complex for the case where they raise degree.

Remark

Frequently one also considers $\mathbb{N}$-graded (or nonnegatively graded) chain complexes (for instance in the Dold-Kan correspondence), which can be identified with $\mathbb{Z}$-graded ones for which $V_n=0$ when $n\lt 0$. Similarly, an $\mathbb{N}$-graded cochain complex is a cochain complex for which $V_n=0$ when $n\lt 0$, or equivalently a chain complex for which $V_n=0$ when $n\gt 0$.

In terms of translations

Note that in particular, a chain complex is a graded object with extra structure. This extra structure can be codified as a map of graded objects $\partial:V\to T V$, where $T$ is the ‘shift’ endofunctor of the category $Gr(C)$ of graded objects in $C$, such that $T(\partial) \circ \partial = 0$. More generally, in any pre-additive category $G$ with translation $T : G \to G$, we can define a chain complex to be a differential object $\partial_V : V \to T V$ such that $V \stackrel{\partial_V}{\to} T V \stackrel{T(\partial_V)}{\to} T T V$ is the zero morphism. When $G= Gr(C)$ this recovers the original definition.

Examples

Common choices for the ambient abelian category $\mathcal{C}$ include Ab, $k$Vect (for $k$ a field) and generally $R$Mod (for $R$ a ring), in which case we obtain the familiar definition of an (unbounded) chain complex of abelian groups, vector spaces and, generally, of modules.

In $k$-vector spaces

In $C =$ Vect$_k$ a chain complex is also called a differential graded vector space, consistent with other terminology such as differential graded algebra over $k$. This is also a special case in other ways: every chain complex over a field is formal: quasi-isomorphic to its homology (the latter considered as a chain complex with zero differentials). Nothing of the sort is true for chain complexes in more general categories.

The category of chain complexes of vector spaces carries the tensor product of chain complexes and a braiding which makes it a symmetric monoidal category. The corresponding commutative monoids are the differential graded-commutative algebras.

In super vector spaces

A chain complex in super vector spaces is a chain complex in super vector spaces. The category of these carries a symmetric monoidal category-structure and the corresponging commutative monoids are the differential graded-commutative superalgebras.

In chain complexes

A chain complex in a category of chain complexes is a double complex.

Singular and cellular chain complex

For $X$ a topological space, there is its singular simplicial complex.

More generally, for $S$ a simplicial set, there is the chain complex $S \cdot R$ of $R$ chains on a simplicial set.

Of a simplicial abelian group

For $A_\bullet$ a simplicial abelian group, there is a chain complex $C_\bullet(A)$, the alternating face map complex, and a chain complex $N_\bullet(A)$, the normalized chain complex of $A$.

The Dold-Kan correspondence says that this construction establishes an equivalence of categories between non-negatively-graded chain complexes and simplicial abelian groups.

Properties

Model structure

There is a model category structure on the category $Ch(A)$ of chain complexes in an abelian category. Its homotopy category is the derived category of $A$. See model structure on chain complexes.

There is another Hurewicz model structure on chain complexes whose homotopy category is the homotopy category of chain complexes.

Related concepts

$H_n = Z_n/B_n$	(chain-)homology	(cochain-)cohomology	$H^n = Z^n/B^n$
$C_n$	chain	cochain	$C^n$
$Z_n \subset C_n$	cycle	cocycle	$Z^n \subset C^n$
$B_n \subset C_n$	boundary	coboundary	$B^n \subset C^n$

• category of chain complexes

  • chain complex, connective chain complex, bounded chain complex

  • chain map, quasi-isomorphism

  • chain homotopy

  • model structure on chain complexes

  • elliptic chain complex

• cochain complex

• filtered chain complex

• perfect chain complex

• homological algebra

• rational stable homotopy theory

• equivariant chain complex

References

• Charles Weibel, section 1.1 An Introduction to Homological Algebra.

• Masaki Kashiwara, Pierre Schapira, Section 11 of: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

• Urs Schreiber, chapter II of Introduction to Homological Algebra