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In [[homological algebra]], the '''mapping cone''' is a construction on a map of [[chain complexes]] inspired by the [[mapping cone|analogous construction in topology]].  In the theory of [[triangulated categories]] it is a kind of combined [[kernel (category theory)|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 [[group (mathematics)|group]]s, [[module (mathematics)|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==
The cone may be defined in the category of [[chain 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>A, B</math> be two complexes, with differentials <math>d_A, d_B;</math> i.e.,
:<math>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>B.</math>


For a map of complexes <math>f : A \to B,</math> we define the cone, often denoted by <math>\operatorname{Cone}(f)</math> or <math>C(f),</math> to be the following complex:
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:<math>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>d_{C(f)} = \begin{pmatrix} d_{A[1]} & 0 \\ f[1] & d_B \end{pmatrix}</math> (acting as though on [[column vector]]s).
Here <math>A[1]</math> is the complex with <math>A[1]^n=A^{n + 1}</math> and <math>d^n_{A[1]}=-d^{n + 1}_{A}</math>.
Note that the differential on <math>C(f)</math> is different from the natural differential on <math>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
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:<math>\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 [[cohomology]] of a complex is defined. The main use of the cone is to identify [[quasi-isomorphism]]s: if the cone is [[acyclic]], then the map is a quasi-isomorphism. To see this, we use the existence of a [[triangulated category#Definition|triangle]]
:<math>A \xrightarrow{f} B \to C(f) \to</math>
where the maps <math>B \to C(f), C(f) \to A[1]</math> are the projections onto the direct summands (see [[Homotopy category of chain complexes]]). Since this is a triangle, it gives rise to a [[long exact sequence]] on [[cohomology group]]s:
:<math>\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>C(f)</math> is acyclic then by definition, the outer terms above are zero. Since the sequence is exact, this means that <math>f^*</math> induces an isomorphism on all cohomology 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>A, B</math> have only one nonzero term in degree 0:
:<math>A = \dots \to 0 \to A^0 \to 0 \to \cdots,</math>
:<math>B = \dots \to 0 \to B^0 \to 0 \to \cdots,</math>
and therefore <math>f \colon A \to B</math> is just <math>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>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 cohomology of this complex is then
:<math>H^{-1}(C(f)) = \operatorname{ker}(f^0),</math>
:<math>H^0(C(f)) = \operatorname{coker}(f^0),</math>
:<math>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 ''f: A  → B'' be a morphism of 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 space]]s <math>\phi : X \rightarrow Y</math>: the complex of [[Singular homology|singular chains]] of the topological cone <math>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==
* {{Citation | last1=Manin | first1=Yuri Ivanovich | author1-link = Yuri Ivanovich Manin | last2=Gelfand | first2=Sergei I. | title=Methods of Homological Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-43583-9 | year=2003}}
* {{Weibel IHA}}
 
{{DEFAULTSORT:Mapping Cone (Homological Algebra)}}
[[Category:Homological algebra]]

Latest revision as of 13:20, 1 May 2014


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