# nLab chain complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## 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 $\{V_n\}_{n \in \mathbb{N}}$ 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

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

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 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 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 $Gr(V)$ 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 $V \stackrel{\partial_V}{\to} T V \stackrel{T(\partial_V)}{\to} T T V$ 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.

### 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 $A$.

$H_n = Z_n/B_n$(chain-)homology(cochain-)cohomology$H^n = Z^n/B^n$
$C_n$chaincochain$C^n$
$Z_n \subset C_n$cyclecocycle$Z^n \subset C^n$
$B_n \subset C_n$boundarycoboundary$B^n \subset C^n$

## References

A basic discussion is for instance in section 1.1 of

A more comprehensive discussion is in section 11 of

Revised on June 25, 2013 04:37:42 by can I even register here? (188.223.15.214)