# 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 ${C}_{•}\left(X\right)$ of a topological space $X$.

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

### Basic

A chain complex ${V}_{•}$ is a sequence $\left\{{V}_{n}{\right\}}_{n\in ℕ}$ of abelian groups or modules (for instance vector spaces) or similar equipped with linear maps $\left\{{d}_{n}:{V}_{n+1}\to {V}_{n}\right\}$ 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

$\mathrm{NCC}:{\mathrm{AB}}^{\Delta }{}^{\mathrm{op}}\to {\mathrm{Ch}}_{•}^{+}\left(\mathrm{AB}\right)$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 $𝒞$ be an abelian category.

###### Definition

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

• a collection of objects $\left\{{C}_{n}{\right\}}_{n\in ℤ}$,

• and of morphisms ${\partial }_{n}:{C}_{n}\to {C}_{n-1}$

$\cdots \stackrel{{\partial }_{3}}{\to }{C}_{2}\stackrel{{\partial }_{2}}{\to }{C}_{1}\stackrel{{\partial }_{1}}{\to }{C}_{0}\stackrel{{\partial }_{0}}{\to }{C}_{-1}\stackrel{{\partial }_{-1}}{\to }\cdots$\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

${\partial }_{n}\circ {\partial }_{n+1}=0$\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 ${\mathrm{Ch}}_{•}\left(𝒞\right)$.

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

${Z}_{n}≔\mathrm{ker}\left({\partial }_{n-1}\right)$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

${B}_{n}≔\mathrm{im}\left({\partial }_{n}\right)$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}\phantom{\rule{thinmathspace}{0ex}}.$0 \hookrightarrow B_n \hookrightarrow Z_n \hookrightarrow 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\phantom{\rule{thinmathspace}{0ex}}.$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 ${𝒞}^{\mathrm{op}}$. 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 $\partial :V\to TV$, where $T$ is the ‘shift’ endofunctor of the category $\mathrm{Gr}\left(V\right)$ of graded objects in $C$, such that $T\left(\partial \right)\circ \partial =0$. 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 ${\partial }_{V}:V\to TV$ such that $V\stackrel{{\partial }_{V}}{\to }TV\stackrel{T\left({\partial }_{V}\right)}{\to }TTV$ is the zero morphism. When $G=\mathrm{Gr}\left(C\right)$ 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 ${C}_{•}\left(A\right)$, the alternating face map complex, and a chain complex ${N}_{•}\left(A\right)$, 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 $\mathrm{Ch}\left(A\right)$ 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 March 6, 2013 20:54:09 by Peter Le Fanu Lumsdaine (192.16.204.218)