and
nonabelian homological algebra
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.
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.
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
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.
Let $\mathcal{C}$ be an abelian category.
A ($\mathbb{Z}$-graded) chain complex in $\mathcal{C}$ is
such that
(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:
For $C_\bullet$ a chain complex
the morphisms $\partial_n$ are called the differentials or boundary maps;
the elements in the kernel
of $\partial_{n-1} : C_n \to C_{n-1}$ are called the $n$-cycles;
the elements in the image
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
the cokernel
is called the degree-$n$ chain homology of $C_\bullet$.
The dual notion:
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.
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.
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 $n\gt 0$.
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.
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 $C =$ 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.
A chain complex in a category of chain complexes is a double 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.
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.
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.
$H_n = Z_n/B_n$ | (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 | $B^n \subset C^n$ |
Charles Weibel, section 1.1 An Introduction to Homological Algebra.
Masaki Kashiwara, Pierre Schapira, section 11 of Categories and Sheaves,
Urs Schreiber, chapter II of Introduction to Homological Algebra