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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.

• Definition

• Properties

• Mapping cylinder

• Topological inspiration

• References

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

A = ⋯ ⋯ → → A n − − 1 → d A n − − 1 A n → d A n A n + 1 → → ⋯ ⋯ {\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 B . {\displaystyle B.}

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

C ( f ) = A [ 1 ] ⊕ ⊕ B = ⋯ ⋯ → → A n ⊕ ⊕ B n − − 1 → → A n + 1 ⊕ ⊕ B n → → A n + 2 ⊕ ⊕ B n + 1 → → ⋯ ⋯ {\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

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

Here A [ 1 ] {\displaystyle A[1]} is the complex with A [ 1 ] n = A n + 1 {\displaystyle A[1]^{n}=A^{n+1}} and d A [ 1 ] n = − − d A n + 1 {\displaystyle d_{A[1]}^{n}=-d_{A}^{n+1}} . Note that the differential on C ( f ) {\displaystyle C(f)} is different from the natural differential on A [ 1 ] ⊕ ⊕ B {\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

d C ( f ) n ( a n + 1 , b n ) = ( d A [ 1 ] n 0 f [ 1 ] n d B n ) ( a n + 1 b n ) = ( − − d A n + 1 0 f n + 1 d B n ) ( a n + 1 b n ) = ( − − d A n + 1 ( a n + 1 ) f n + 1 ( a n + 1 ) + d B n ( b n ) ) = ( − − d A n + 1 ( a n + 1 ) , f n + 1 ( a n + 1 ) + d B n ( b n ) ) . {\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

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

where the maps B → → C ( f ) , C ( f ) → → A [ 1 ] {\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:

⋯ ⋯ → → H i − − 1 ( C ( f ) ) → → H i ( A ) → f ∗ ∗ H i ( B ) → → H i ( C ( f ) ) → → ⋯ ⋯ {\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 C ( f ) {\displaystyle C(f)} is acyclic then by definition, the outer terms above are zero. Since the sequence is exact, this means that f ∗ ∗ {\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 A , B {\displaystyle A,B} have only one nonzero term in degree 0:

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

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

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

C ( f ) = ⋯ ⋯ → → 0 → → A 0 [ − − 1 ] → f 0 B 0 [ 0 ] → → 0 → → ⋯ ⋯ . {\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

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

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

H i ( C ( f ) ) = 0 for i ≠ ≠ − − 1 , 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 f : : A → → B {\displaystyle f\colon A\to B} be a morphism of chain complexes, let further g : : Cone ⁡ ⁡ ( f ) [ − − 1 ] → → A {\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 ϕ ϕ : X → → Y {\displaystyle \phi :X\rightarrow Y} : the complex of singular chains of the topological cone c o n e ( ϕ ϕ ) {\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.

• citerefmaningelfand2003Manin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43583-9

• citerefweibel1994Weibel, 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)