chain complex in nLab

Context

Homological algebra

Idea
- Basic
- Meaning in homotopy theory
Definition
- In components
- In terms of translations
Examples
Properties
- Model structure
Related concepts
References

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 $C_{•} (X)$ of a topological space $X$ .

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

Basic

A chain complex $V_{•}$ is a sequence ${V_{n}}_{n \in ℕ}$ 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.

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}^{Δ}^{op} \to {Ch}_{•}^{+} (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 $𝒞$ be an abelian category.

Definition

A ( $ℤ$ -graded) chain complex in $𝒞$ is

a collection of objects ${C_{n}}_{n \in ℤ}$ ,
and of morphisms $\partial_{n} : C_{n} \to C_{n - 1}$

\dots \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} \dots

such that

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

(the zero morphism) for all $n \in ℕ$ .

A homomorphism of chain complexes is a chain map (see there). Chain complexes with chain maps between them form the category of chain complexes ${Ch}_{•} (𝒞)$ .

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

Definition

For $C_{•}$ a chain complex

the morphisms $\partial_{n}$ are called the differentials or boundary maps;
the elements of are called the $n$ -chains;
the elements in the kernel

$Z_{n} ≔ \ker (\partial_{n - 1})$ 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

$B_{n} ≔ im (\partial_{n})$ 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 ↪ B_{n} ↪ Z_{n} ↪ C_{n} .

the cokernel

$H_{n} ≔ Z_{n} / B_{n}$ H_n \coloneqq Z_n/B_n

is called the degree- $n$ chain homology of $C_{•}$ .

0 \to B_{n} \to Z_{n} \to H_{n} \to 0 .

The dual notion:

Definition

A cochain complex in $𝒞$ is a chain complex in the opposite category $𝒞^{op}$ . Hence a tower of 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 $ℕ$ -graded (or nonnegatively graded) chain complexes (for instance in the Dold-Kan correspondence), which can be identified with $ℤ$ -graded ones for which $V_{n} = 0$ when $n < 0$ . Similarly, an $ℕ$ -graded cochain complex is a cochain complex for which $V_{n} = 0$ when $n < 0$ , or equivalently a chain complex for which $V_{n} = 0$ when $n > 0$ .

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 $Gr (V)$ 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 $V \overset{\partial_{V}}{\to} T V \overset{T (\partial_{V})}{\to} T T V$ is the zero morphism. When $G = Gr (C)$ this recovers the original definition.

Examples

Common choices for the ambient abelian category $𝒞$ 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.

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 $A_{•}$ a simplicial abelian group, there is a chain complex $C_{•} (A)$ , the alternating face map complex, and a chain complex $N_{•} (A)$ , 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.

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	$B^{n} \subset C^{n}$

References

A basic discussion is for instance in section 1.1 of

Charles Weibel, An Introduction to Homological Algebra.

A more comprehensive discussion is in section 11 of

Masaki Kashiwara, Pierre Schapira, Categories and Sheaves,

nLab
chain complex

Context

Homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

Homology theories

Theorems

Contents

Idea

Basic

Meaning in homotopy theory

Definition

In components

Definition

Definition

Definition

Remark

Remark

In terms of translations

Examples

In $k$ -vector spaces

In chain complexes

Singular and cellular chain complex

Of a simplicial abelian group

Properties

Model structure

Related concepts

References

nLab chain complex

Context

Homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

Homology theories

Theorems

Contents

Idea

Basic

Meaning in homotopy theory

Definition

In components

Definition

Definition

Definition

Remark

Remark

In terms of translations

Examples

In k-vector spaces

In chain complexes

Singular and cellular chain complex

Of a simplicial abelian group

Properties

Model structure

Related concepts

References

nLab
chain complex

In $k$ -vector spaces