Mapping cone (homological algebra)

Short description: Tool in homological algebra

In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi-isomorphism; if we pass to the derived category of complexes, this means that f is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism. If we are working in a t-category, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.

## Contents

* 1 Definition
* 2 Properties
* 3 Mapping cylinder
* 4 Topological inspiration
* 5 References

## Definition

The cone may be defined in the category of cochain complexes over any additive category (i.e., a category whose morphisms form abelian groups and in which we may construct a direct sum of any two objects). Let [math]\displaystyle{ A, B }[/math] be two complexes, with differentials [math]\displaystyle{ d_A, d_B; }[/math] i.e.,

[math]\displaystyle{ A = \dots \to A^{n - 1} \xrightarrow{d_A^{n - 1}} A^n \xrightarrow{d_A^n} A^{n + 1} \to \cdots }[/math]

and likewise for [math]\displaystyle{ B. }[/math]

For a map of complexes [math]\displaystyle{ f : A \to B, }[/math] we define the cone, often denoted by [math]\displaystyle{ \operatorname{Cone}(f) }[/math] or [math]\displaystyle{ C(f), }[/math] to be the following complex:

[math]\displaystyle{ C(f) = A[1] \oplus B = \dots \to A^n \oplus B^{n - 1} \to A^{n + 1} \oplus B^n \to A^{n + 2} \oplus B^{n + 1} \to \cdots }[/math] on terms,

with differential

[math]\displaystyle{ d_{C(f)} = \begin{pmatrix} d_{A[1]} & 0 \\\ f[1] & d_B \end{pmatrix} }[/math] (acting as though on column vectors).

Here [math]\displaystyle{ A[1] }[/math] is the complex with [math]\displaystyle{ A[1]^n=A^{n + 1} }[/math] and [math]\displaystyle{ d^n_{A[1]}=-d^{n + 1}_{A} }[/math]. Note that the differential on [math]\displaystyle{ C(f) }[/math] is different from the natural differential on [math]\displaystyle{ A[1] \oplus B }[/math], and that some authors use a different sign convention.

Thus, if for example our complexes are of abelian groups, the differential would act as

[math]\displaystyle{ \begin{array}{ccl} d^n_{C(f)}(a^{n + 1}, b^n) &=& \begin{pmatrix} d^n_{A[1]} & 0 \\\ f[1]^n & d^n_B \end{pmatrix} \begin{pmatrix} a^{n + 1} \\\ b^n \end{pmatrix} \\\ &=& \begin{pmatrix} - d^{n + 1}_A & 0 \\\ f^{n + 1} & d^n_B \end{pmatrix} \begin{pmatrix} a^{n + 1} \\\ b^n \end{pmatrix} \\\ &=& \begin{pmatrix} - d^{n + 1}_A (a^{n + 1}) \\\ f^{n + 1}(a^{n + 1}) + d^n_B(b^n) \end{pmatrix}\\\ &=& \left(- d^{n + 1}_A (a^{n + 1}), f^{n + 1}(a^{n + 1}) + d^n_B(b^n)\right). \end{array} }[/math]

## Properties

Suppose now that we are working over an abelian category, so that the homology of a complex is defined. The main use of the cone is to identify quasi-isomorphisms: if the cone is acyclic, then the map is a quasi-isomorphism. To see this, we use the existence of a triangle

[math]\displaystyle{ A \xrightarrow{f} B \to C(f) \to A[1] }[/math]

where the maps [math]\displaystyle{ B \to C(f), C(f) \to A[1] }[/math] are given by the direct summands (see Homotopy category of chain complexes). Since this is a triangle, it gives rise to a long exact sequence on homology groups:

[math]\displaystyle{ \dots \to H_{i - 1}(C(f)) \to H_i(A) \xrightarrow{f^*} H_i(B) \to H_i(C(f)) \to \cdots }[/math]

and if [math]\displaystyle{ C(f) }[/math] is acyclic then by definition, the outer terms above are zero. Since the sequence is exact, this means that [math]\displaystyle{ f^* }[/math] induces an isomorphism on all homology groups, and hence (again by definition) is a quasi-isomorphism.

This fact recalls the usual alternative characterization of isomorphisms in an abelian category as those maps whose kernel and cokernel both vanish. This appearance of a cone as a combined kernel and cokernel is not accidental; in fact, under certain circumstances the cone literally embodies both. Say for example that we are working over an abelian category and [math]\displaystyle{ A, B }[/math] have only one nonzero term in degree 0:

[math]\displaystyle{ A = \dots \to 0 \to A_0 \to 0 \to \cdots, }[/math]
[math]\displaystyle{ B = \dots \to 0 \to B_0 \to 0 \to \cdots, }[/math]

and therefore [math]\displaystyle{ f \colon A \to B }[/math] is just [math]\displaystyle{ f_0 \colon A_0 \to B_0 }[/math] (as a map of objects of the underlying abelian category). Then the cone is just

[math]\displaystyle{ C(f) = \dots \to 0 \to \underset{[-1]}{A_0} \xrightarrow{f_0} \underset{[0]}{B_0} \to 0 \to \cdots. }[/math]

(Underset text indicates the degree of each term.) The homology of this complex is then

[math]\displaystyle{ H_{-1}(C(f)) = \operatorname{ker}(f_0), }[/math]
[math]\displaystyle{ H_0(C(f)) = \operatorname{coker}(f_0), }[/math]
[math]\displaystyle{ H_i(C(f)) = 0 \text{ for } i \neq -1, 0.\ }[/math]

This is not an accident and in fact occurs in every t-category.

## Mapping cylinder

A related notion is the mapping cylinder: let [math]\displaystyle{ f\colon A \to B }[/math] be a morphism of chain complexes, let further [math]\displaystyle{ g \colon \operatorname{Cone}(f)[-1] \to A }[/math] be the natural map. The mapping cylinder of f is by definition the mapping cone of g.

## Topological inspiration

This complex is called the cone in analogy to the mapping cone (topology) of a continuous map of topological spaces [math]\displaystyle{ \phi : X \rightarrow Y }[/math]: the complex of singular chains of the topological cone [math]\displaystyle{ cone(\phi) }[/math] is homotopy equivalent to the cone (in the chain-complex-sense) of the induced map of singular chains of X to Y. The mapping cylinder of a map of complexes is similarly related to the mapping cylinder of continuous maps.

## References

* Manin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43583-9
* Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. OCLC 36131259.
* Joeseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See chapter 9)

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