Mapping cone (homological algebra)

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 ${\displaystyle A,B}$ be two complexes, with differentials ${\displaystyle d_{A},d_{B};}$ i.e.,

${\displaystyle A=\dots \to A^{n-1}{\xrightarrow {d_{A}^{n-1}}}A^{n}{\xrightarrow {d_{A}^{n}}}A^{n+1}\to \cdots }$

and likewise for ${\displaystyle B.}$

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

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

with differential

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

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

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

${\displaystyle {\begin{array}{ccl}d_{C(f)}^{n}(a^{n+1},b^{n})&=&{\begin{pmatrix}d_{A[1]}^{n}&0\\f[1]^{n}&d_{B}^{n}\end{pmatrix}}{\begin{pmatrix}a^{n+1}\\b^{n}\end{pmatrix}}\\&=&{\begin{pmatrix}-d_{A}^{n+1}&0\\f^{n+1}&d_{B}^{n}\end{pmatrix}}{\begin{pmatrix}a^{n+1}\\b^{n}\end{pmatrix}}\\&=&{\begin{pmatrix}-d_{A}^{n+1}(a^{n+1})\\f^{n+1}(a^{n+1})+d_{B}^{n}(b^{n})\end{pmatrix}}\\&=&\left(-d_{A}^{n+1}(a^{n+1}),f^{n+1}(a^{n+1})+d_{B}^{n}(b^{n})\right).\end{array}}}$

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

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

where the maps ${\displaystyle B\to C(f),C(f)\to A[1]}$ 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:

${\displaystyle \dots \to H_{i-1}(C(f))\to H_{i}(A){\xrightarrow {f^{*}}}H_{i}(B)\to H_{i}(C(f))\to \cdots }$

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

${\displaystyle A=\dots \to 0\to A_{0}\to 0\to \cdots ,}$

${\displaystyle B=\dots \to 0\to B_{0}\to 0\to \cdots ,}$

and therefore ${\displaystyle f\colon A\to B}$ is just ${\displaystyle f_{0}\colon A_{0}\to B_{0}}$ (as a map of objects of the underlying abelian category). Then the cone is just

${\displaystyle C(f)=\dots \to 0\to {\underset {[-1]}{A_{0}}}{\xrightarrow {f_{0}}}{\underset {[0]}{B_{0}}}\to 0\to \cdots .}$

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

${\displaystyle H_{-1}(C(f))=\operatorname {ker} (f_{0}),}$

${\displaystyle H_{0}(C(f))=\operatorname {coker} (f_{0}),}$

${\displaystyle H_{i}(C(f))=0{\text{ for }}i\neq -1,0.\ }$

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

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

This complex is called the cone in analogy to the mapping cone (topology) of a continuous map of topological spaces ${\displaystyle \phi :X\rightarrow Y}$: the complex of singular chains of the topological cone ${\displaystyle cone(\phi )}$ 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.

• 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. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
• Joeseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See chapter 9)