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 A,B be two complexes, with differentials dA,dB; i.e.,

A=…->An−1→dAn−1An→dAnAn+1->⋯

and likewise for B.

For a map of complexes f:A->B, we define the cone, often denoted by Cone⁡(f) or C(f), to be the following complex:

C(f)=A[1]⊕B=…->An⊕Bn−1->An+1⊕Bn->An+2⊕Bn+1->⋯ on terms,

with differential

dC(f)=(dA[1]0f[1]dB) (acting as though on column vectors).

Here A[1] is the complex with A[1]n=An+1 and dA[1]n=−dAn+1. Note that the differential on C(f) is different from the natural differential on A[1]⊕B, and that some authors use a different sign convention.

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

dC(f)n(an+1,bn)=(dA[1]n0f[1]ndBn)(an+1bn)=(−dAn+10fn+1dBn)(an+1bn)=(−dAn+1(an+1)fn+1(an+1)+dBn(bn))=(−dAn+1(an+1),fn+1(an+1)+dBn(bn)).

## 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

A→fB->C(f)->A[1]

where the maps B->C(f),C(f)->A[1] 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:

…->Hi−1(C(f))->Hi(A)→f*Hi(B)->Hi(C(f))->⋯

and if C(f) is acyclic then by definition, the outer terms above are zero. Since the sequence is exact, this means that f* 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 A,B have only one nonzero term in degree 0:

A=…->0->A0->0->⋯,
B=…->0->B0->0->⋯,

and therefore f:A->B is just f0:A0->B0 (as a map of objects of the underlying abelian category). Then the cone is just

C(f)=…->0->A0[−1]→f0B0[0]->0->⋯.

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

H−1(C(f))=ker⁡(f0),
H0(C(f))=coker⁡(f0),
Hi(C(f))=0 for i≠−1,0.

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

## Mapping cylinder

A related notion is the mapping cylinder: let f:A->B be a morphism of chain complexes, let further g:Cone⁡(f)[−1]->A 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 ϕ:X->Y: the complex of singular chains of the topological cone cone(ϕ) 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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