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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{mapping cone} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits} [[!include infinity-limits - contents]] \hypertarget{homotopy}{}\paragraph*{{Homotopy}}\label{homotopy} [[!include homotopy - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{suspension}{Suspension}\dotfill \pageref*{suspension} \linebreak \noindent\hyperlink{ForTopologicalSpaces}{For topological spaces}\dotfill \pageref*{ForTopologicalSpaces} \linebreak \noindent\hyperlink{InChainComplexes}{For chain complexes}\dotfill \pageref*{InChainComplexes} \linebreak \noindent\hyperlink{InCochainComplexes}{For cochain complexes}\dotfill \pageref*{InCochainComplexes} \linebreak \noindent\hyperlink{in_additive_categories_with_translation}{In additive categories with translation}\dotfill \pageref*{in_additive_categories_with_translation} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{HomologyExactSequencesAndFiberSequences}{Homology exact sequences and fiber sequences}\dotfill \pageref*{HomologyExactSequencesAndFiberSequences} \linebreak \noindent\hyperlink{DistinguishedTriangles}{Distinguished triangles from mapping cones}\dotfill \pageref*{DistinguishedTriangles} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{mapping cone} of a [[morphism]] $f : X \to Y$ in some [[homotopical category]] (precisely: a [[category of cofibrant objects]]) is, if it exists, a particular representative of the [[homotopy fiber|homotopy cofiber]] of $f$. It is also called the \emph{homotopy [[cokernel]]} of $f$ or the \emph{[[weak quotient]]} of $Y$ by the [[image]] of $X$ in $Y$ under $f$. The dual notion is that of [[mapping cocone]]. (graphics taken from \href{http://personal.us.es/fmuro/praha.pdf}{Muro 10}) \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} The mapping cone construction is a means to \emph{present} in a [[category with weak equivalences]] the following canonical construction in [[homotopy theory]]/[[(∞,1)-category theory]]. the \begin{defn} \label{InfinityCokernel}\hypertarget{InfinityCokernel}{} In an [[(∞,1)-category]] $\mathcal{C}$ with [[terminal object]] and [[(∞,1)-pushout]], the [[homotopy fiber|homotopy cofiber]] of a [[morphism]] $f : X \to Y$ is the [[homotopy pushout]] \begin{displaymath} coker(f) \coloneqq Y \coprod_X {*} \end{displaymath} hence the object [[universal construction]] sitting universally in a [[diagram]] of the form \begin{displaymath} \itexarray{ X &\stackrel{}{\to}& {*} \\ \downarrow^{\mathrlap{f}} &\swArrow_{\simeq}& \downarrow \\ Y &\to& coker(f) } \,. \end{displaymath} \end{defn} \begin{prop} \label{HomotopyCofiberByFactorizationLemma}\hypertarget{HomotopyCofiberByFactorizationLemma}{} If the [[(∞,1)-category]] $\mathcal{C}$ is presented by (is [[equivalence of (infinity,1)-categories|equivalent]] to the [[simplicial localization]] of) a [[category of cofibrant objects]] $C$ (for instance given by the [[cofibrant objects]] in a [[model category]]) then this homotopy cofiber is presented by the ordinary [[colimit]] \begin{displaymath} \itexarray{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\to}& cyl(X) \\ \downarrow && &\searrow & \downarrow \\ {*} &\to& &\to& cone(f) } \end{displaymath} in $C$ using any [[cylinder object]] $cyl(X)$ for $X$. \end{prop} This is discussed in detail at [[factorization lemma]] and at [[homotopy pullback]]. \begin{remark} \label{}\hypertarget{}{} Intuitively this says that $cone(f)$ is the object obtained by \begin{enumerate}% \item forming the cylinder over $X$; \item gluing to one end of that the object $Y$ as specified by the map $f$. \item shrinking the other end of the cylinder to the point. \end{enumerate} Intuitively it is clear that this way every [[cycle]] in $Y$ that happens to be in the image of $X$ can be ``continuously'' translated in the cylinder-direction, keeping it constant in $Y$, to the other end of the cylinder, where it becomes the point. This means that every [[homotopy group]] of $Y$ in the image of $f$ vanishes in the mapping cone. Hence in the mapping conee \textbf{the image of $X$ under $f$ in $Y$ is removed up to homotopy}. This makes it clear how $cone(f)$ is a homotopy-version of the [[cokernel]] of $f$. And therefore the name ``mapping cone''. \end{remark} \begin{remark} \label{}\hypertarget{}{} A morphism $\eta : cyl(X) \to Y$ out of a [[cylinder object]] is a [[left homotopy]] $\eta : g \Rightarrow h$ between its restrictions $g\coloneqq \eta(0)$ and $h \coloneqq \eta(1)$ to the cylinder boundaries \begin{displaymath} \itexarray{ X \\ \downarrow^{\mathrlap{i_0}} & \searrow^{\mathrlap{g}} \\ cyl(X) &\stackrel{\eta}{\to}& Y \\ \uparrow^{\mathrlap{i_1}} & \nearrow_{\mathrlap{h}} \\ X } \,. \end{displaymath} Therefore prop. \ref{HomotopyCofiberByFactorizationLemma} says that the mapping cone is the the [[universal property|universal]] object with a morphism $i$ from $Y$ and a [[left homotopy]] from $i \circ f$ to the [[zero morphism]]. This is of course also precisely what def. \ref{InfinityCokernel} is saying. \end{remark} \begin{prop} \label{}\hypertarget{}{} The colimit in prop. \ref{HomotopyCofiberByFactorizationLemma} may be computed in two stages by two consecutive [[pushouts]] in $C$, and in two ways by the following [[pasting diagram]]: \begin{displaymath} \itexarray{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{i_1} && \downarrow \\ X &\stackrel{i_0}{\to}& cyl(X) &\to & cyl(f) \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& cone(X) &\to& cone(f) } \,. \end{displaymath} Here every square is a [[pushout]], (and so by the [[pasting law]] is every rectangular pasting composite). \end{prop} This now is a basic fact in ordinary [[category theory]]. The pushouts appearing here go by the following names: \begin{defn} \label{CylindersAndCones}\hypertarget{CylindersAndCones}{} The pushout \begin{displaymath} \itexarray{ X &\stackrel{i_0}{\to}& cyl(X) \\ \downarrow && \downarrow \\ {*} &\to& cone(X) } \end{displaymath} defines the \textbf{[[cone]]} $cone(X)$ over $X$ (with respect to the chosen [[cylinder object]]): the result of taking the [[cylinder object|cylinder]] over $X$ and identifying one $X$-shaped end with the [[point]]. The pushout \begin{displaymath} \itexarray{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow \\ cyl(X) &\to& cyl(f) } \end{displaymath} defines the \textbf{[[mapping cylinder]]} $cyl(f)$ of $f$, the result of identifying one end of the cylinder over $X$ with $Y$, using $f$ as the gluing map. The pushout \begin{displaymath} \itexarray{ cyl(x) &\to& cyl(f) \\ \downarrow && \downarrow \\ cone(X) &\to& cone(f) } \end{displaymath} defines the \textbf{mapping cone} $cone(f)$ of $f$: the result of forming the cylinder over $X$ and then identifying one end with the point and the other with $Y$, via $f$. \end{defn} The geometric intuition behind this is best seen in the archetypical example of the [[classical model structure on topological spaces]]. See the example \emph{\hyperlink{ForTopologicalSpaces}{For topological spaces}} below. The example \emph{\href{InChainComplexes}{For chain complexes}} can be understood similarly geometrically by thinking of all chain complexes as [[singular chain complex|singular chains]] on topological spaces. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} We discuss realizations of the general construction in various contexts. Some of these examples are regarded in parts of the literature as the default examples, notably that \hyperlink{ForTopologicalSpaces}{for topological spaces} and that \hyperlink{InChainComplexes}{for chain complexes}. \hypertarget{suspension}{}\subsubsection*{{Suspension}}\label{suspension} The mapping cone of the morphism $X \to {*}$ to the [[terminal object]] is the [[suspension object]] $\Sigma X$ of an object $X$. The dual notion of the [[loop space object]] of $X$. \hypertarget{ForTopologicalSpaces}{}\subsubsection*{{For topological spaces}}\label{ForTopologicalSpaces} The notion \emph{mapping cone} derives its name from its geometrical interpretation in the category [[Top]] of [[topological spaces]]. For more details see also at \emph{[[topological cofiber sequence]]}. With respect to the standard [[model structure on topological spaces]] every [[CW-complex]] is a cofibrant object, and hence mapping cones on maps between CW-complexes have intrinsic meaning in [[homotopy theory]]. Write $I \coloneqq [0,1] \subset \mathbb{R} \in$ [[Top]] for the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]]. This is an [[interval object]] for the standard model structure. We may therefore take the [[cylinder object]] of a topological space $X$ to be \begin{displaymath} cyl(X) \coloneqq X \times I \,, \end{displaymath} which is literally the cylinder over $X$. Given a [[continuous function]] $f:X\to Y$, the [[topological space]] $cone(f)$ is \begin{displaymath} cone(f) = (X \times I) \cup_{f} Y \end{displaymath} This is the [[disjoint union]] of $X \times I$ with $Y$ followed by an identification under which for each $x\in X$ a point $(x,1) \in X \times I$ is identified with the point $f(x) \in Y$ and followed by the contraction of $X\times \{0\}$ to a point. Of course the opposite convention is also possible: identify $(x,0)$ with $f(x)$ for all $x$ and then contract $X\times\{1\}$ to a point; the two constructions of cones are canonically homeomorphic; the first is sometimes called the ``inverse mapping cone''. The [[singular chain complex]] [[functor]] from [[Top]] to the [[category of chain complexes]] of [[abelian groups]] sends the mapping cone to a mapping cone in the sense of chain complexes (up to conventions on the orientation of the interval and vector order in the definition of mapping cone of chain complexes). \hypertarget{InChainComplexes}{}\subsubsection*{{For chain complexes}}\label{InChainComplexes} Let $Ch_\bullet = Ch_\bullet(R Mod)$ be the [[category of chain complexes]] in $R$[[Mod]] for some [[ring]] $R$. (For instance if $R = \mathbb{Z}$ the [[integers]], then this is $Ch_\bullet(Ab)$, chain complexes of [[abelian groups]]. More generally $R Mod$ can be replaced by any [[abelian category]] in the following, with the evident changes in the presentation here and there.) We derive an explicit presentation of the mapping cone $cone(f)$ of a [[chain map]] $f$, according to the general definition \ref{CylindersAndCones}. The end result is prop. \ref{ComponentsOfMappingConeInChainComplexes} below, reproducing the classical formula for the mapping cone. \begin{defn} \label{}\hypertarget{}{} Write $*_\bullet \in Ch_\bullet(\mathcal{A})$ for the chain complex concentrated on $R$ in degree 0 \begin{displaymath} *_\bullet 0 = [\cdots \to 0 \to 0 \to R] \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} This may be understood as the [[normalized chain complex]] of [[simplicial homology|chains of simplices]] on the terminal [[simplicial set]] $\Delta^0$, the 0-[[simplex]]. \end{remark} \begin{defn} \label{BoundaryInclusionIntoChainComplexIntervalObject}\hypertarget{BoundaryInclusionIntoChainComplexIntervalObject}{} Let $I_\bullet \in Ch_{\bullet}(\mathcal{A})$ be given by \begin{displaymath} I_\bullet = (\cdots 0 \to 0 \to R \stackrel{(-id,id)}{\to} R \oplus R) \,. \end{displaymath} Denote by \begin{displaymath} i_0 : *_\bullet \to I_\bullet \end{displaymath} the [[chain map]] which in degree 0 is the canonical inclusion into the second summand of a [[direct sum]] and by \begin{displaymath} i_1 : *_\bullet \to I_\bullet \end{displaymath} correspondingly the canonical inclusion into the first summand. \end{defn} \begin{remark} \label{}\hypertarget{}{} This is the standard [[interval object in chain complexes]]. It is in fact the [[normalized chain complex]] of [[chains on a simplicial set]] for the canonical simplicial interval, the 1-[[simplex]]: \begin{displaymath} I_\bullet = C_\bullet(\Delta[1]) \,. \end{displaymath} The [[differential]] $\partial^I = (-id, id)$ here expresses the [[alternating face map complex]] [[boundary]] operator, which in terms of the three non-degenerate [[basis]] elements is given by \begin{displaymath} \partial ( 0 \to 1 ) = (1) - (0) \,. \end{displaymath} \end{remark} We decompose the proof of this statement is a sequence of substatements. \begin{prop} \label{}\hypertarget{}{} For $X_\bullet \in Ch_\bullet$ the [[tensor product of chain complexes]] \begin{displaymath} (I \otimes X)_\bullet \in Ch_\bullet \end{displaymath} is a [[cylinder object]] of $X_\bullet$ for the structure of a [[category of cofibrant objects]] on $Ch_\bullet$ whose cofibrations are the [[monomorphisms]] and whose weak equivalences are the [[quasi-isomorphisms]] (the substructure of the standard [[injective model structure on chain complexes]]). \end{prop} \begin{prop} \label{}\hypertarget{}{} The complex $(I \otimes X)_\bullet$ has components \begin{displaymath} (I \otimes X)_n = X_n \oplus X_n \oplus X_{n-1} \end{displaymath} and the [[differential]] is given by \begin{displaymath} \itexarray{ X_{n+1} \oplus X_{n+1} &\stackrel{\partial^X \oplus \partial^X}{\to}& X_n \oplus X_n \\ \oplus &\nearrow_{(-id,id)}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,, \end{displaymath} hence in [[matrix calculus]] by \begin{displaymath} \partial^{I \otimes X} = \left( \itexarray{ \partial^X \oplus \partial^X & (-id, id) \\ 0 & -\partial^X } \right) : (X_{n+1} \oplus X_{n+1}) \oplus X_{n} \to (X_{n} \oplus X_{n}) \oplus X_{n-1} \,. \end{displaymath} \end{prop} \begin{proof} By the formula discussed at [[tensor product of chain complexes]] the components arise as the [[direct sum]] \begin{displaymath} (I \otimes X )_n = (R_{(0)} \otimes X_n ) \oplus (R_{(1)} \otimes X_n ) \oplus (R_{(0 \to 1)} \otimes X_{(n-1)} ) \end{displaymath} and the [[differential]] picks up a sign when passed past the degree-1 term $R_{(0 \to 1)}$: \begin{displaymath} \begin{aligned} \partial^{I \otimes X} ( (0 \to 1), x ) &= ( (\partial^I (0 \to 1)), x ) - ( (0\to 1), \partial^X x ) \\ & = ( - (0) + (1), x ) - ( (0 \to 1), \partial^X x ) \\ & = -((0), x) + ((1), x) - ( (0 \to 1), \partial^X x ) \end{aligned} \,. \end{displaymath} \end{proof} \begin{remark} \label{InclusionOfChainComplexIntoItsCylinder}\hypertarget{InclusionOfChainComplexIntoItsCylinder}{} The two boundary inclusions of $X_\bullet$ into the cylinder are given in terms of def. \ref{BoundaryInclusionIntoChainComplexIntervalObject} by \begin{displaymath} i^X_0 : X_\bullet \simeq *_\bullet \otimes X_\bullet \stackrel{i_0 \otimes id_X}{\to} (I\otimes X)_\bullet \end{displaymath} and \begin{displaymath} i^X_1 : X_\bullet \simeq *_\bullet \otimes X_\bullet \stackrel{i_1 \otimes id_X}{\to} (I\otimes X)_\bullet \end{displaymath} which in components is the inclusion of the second or first direct summand, respectively \begin{displaymath} X_n \hookrightarrow X_n \oplus X_n \oplus X_{n-1} \,. \end{displaymath} \end{remark} One part of definition \ref{CylindersAndCones} now reads: \begin{defn} \label{}\hypertarget{}{} For $f_\bullet : X_\bullet \to Y_\bullet$ a [[chain map]], the [[mapping cylinder]] $cyl(f)$ is the [[pushout]] \begin{displaymath} \itexarray{ cyl(f)_\bullet &\leftarrow& Y_\bullet \\ \uparrow && \uparrow^{\mathrlap{f}} \\ I_\bullet \otimes X_\bullet &\stackrel{i_0}{\leftarrow}& X_\bullet } \,. \end{displaymath} \end{defn} \begin{prop} \label{MappingConeForChainComplexesExplicitly}\hypertarget{MappingConeForChainComplexesExplicitly}{} The components of $cyl(f)_\bullet$ are \begin{displaymath} cyl(f)_n = X_n \oplus Y_n \oplus X_{n-1} \end{displaymath} and the [[differential]] is given by \begin{displaymath} \itexarray{ X_{n+1} \oplus Y_{n+1} &\stackrel{\partial^X \oplus \partial^Y}{\to}& X_n \oplus Y_n \\ \oplus &\nearrow_{(-id,f)}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,, \end{displaymath} hence in [[matrix calculus]] by \begin{displaymath} \partial^{cyl(f)} = \left( \itexarray{ \partial^X \oplus \partial^Y & (-id, f_n) \\ 0 & -\partial^X } \right) : (X_{n+1} \oplus Y_{n+1}) \oplus X_{n} \to (X_{n} \oplus Y_{n}) \oplus X_{n-1} \,. \end{displaymath} \end{prop} \begin{proof} The [[colimits]] in a [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$ are computed in the underlying [[presheaf category]] of [[towers]] in $\mathcal{A}$. There they are computed degreewise in $\mathcal{A}$ (see at \href{limits+and+colimits+by+example#limitsinpresheafcat}{limits in presheaf categories}). Here the statement is evident: the pushout identifies one [[direct sum|direct summand]] $X_n$ with $Y_n$ along $f_n$ and so where previously a $id_{X_n}$ appeared on the diagonl, there is now $f_n$. \end{proof} The last part of definition \ref{CylindersAndCones} now reads: \begin{defn} \label{}\hypertarget{}{} For $f_\bullet : X_\bullet \to Y_\bullet$ a [[chain map]], the \textbf{mapping cone} $cone(f)$ is the [[pushout]] \begin{displaymath} \itexarray{ cone(f) &\leftarrow& cyl(f) \\ \uparrow && \uparrow \\ cone(X) &\leftarrow& X \otimes I \\ \uparrow && \uparrow^{\mathrlap{i_1}} \\ 0 &\leftarrow& X } \end{displaymath} \end{defn} In the literature this appears for instance as (\hyperlink{Schapira}{Schapira, def. 3.2.2}). \begin{prop} \label{ComponentsOfMappingConeInChainComplexes}\hypertarget{ComponentsOfMappingConeInChainComplexes}{} The components of the mapping cone $cone(f)$ are \begin{displaymath} cone(f)_n = Y_n \oplus X_{n-1} \end{displaymath} with differential given by \begin{displaymath} \itexarray{ Y_{n+1} &\stackrel{\partial^Y}{\to}& Y_n \\ \oplus &\nearrow_{f_n}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,, \end{displaymath} and hence in [[matrix calculus]] by \begin{displaymath} \partial^{cone(f)} = \left( \itexarray{ \partial^Y_{n+1} & f_n \\ 0 & -\partial^X_n } \right) : Y_{n+1} \oplus X_{n} \to Y_{n} \oplus X_{n-1} \,. \end{displaymath} \end{prop} \begin{proof} As before the pushout is computed degreewise. This identifies the remaining unshifted copy of $X$ with 0. \end{proof} \begin{prop} \label{InclusionIntoChainComplexCone}\hypertarget{InclusionIntoChainComplexCone}{} For $f : X_\bullet \to Y_\bullet$ a [[chain map]], the canonical inclusion $i : Y_\bullet \to cone(f)_\bullet$ of $Y_\bullet$ into the mapping cone of $f$ is given in components \begin{displaymath} i_n : Y_n \to cone(f)_n = Y_n \oplus X_{n-1} \end{displaymath} by the canonical inclusion of a summand into a [[direct sum]]. \end{prop} \begin{proof} This follows by starting with remark \ref{InclusionOfChainComplexIntoItsCylinder} and then following these inclusions through the formation of the two colimits as discussed above. \end{proof} The construction above builds the mapping cone explicitly via the standard formula for [[homotopy pushouts]]. Often however other presentations are more convenient: \begin{prop} \label{ConeViaDoubleComplex}\hypertarget{ConeViaDoubleComplex}{} For $f_\bullet \colon X_\bullet \to Y_\bullet$ a [[chain map]], consider the [[double complex]] $D_{\bullet,\bullet}$ concentrated in degrees $D_{1,\bullet} \coloneqq X_\bullet$ and $D_{0,\bullet} \coloneqq Y_\bullet$ with $\partial_{0,\bullet} \coloneqq f_\bullet \colon D_{1,\bullet} \to D_{0,\bullet}$. Then the [[total complex]] of $D_{\bullet, \bullet}$ is also a model for the mapping cone of $f$: \begin{displaymath} Cone(f) \simeq tot(D_{\bullet,\bullet}) \,. \end{displaymath} \end{prop} \begin{proof} One checks by inspection that $tot(D_{\bullet,\bullet}) = Cone(\tilde f)$ for $\tilde f\colon X_\bullet \to Y_\bullet$ for which there is a [[chain homotopy]] $f \Rightarrow f'$ (given only by multiplication by signs). \end{proof} This appears for instance as (\hyperlink{Weibel}{Weibel, Exercise 1.2.8}). \hypertarget{InCochainComplexes}{}\subsubsection*{{For cochain complexes}}\label{InCochainComplexes} We spell out the situation in more detail in a [[category of cochain complexes]]. Let $\mathcal{A}$ be some [[concrete category|concrete]] [[additive category]] and $Ch^\bullet(\mathcal{A})$ the [[category of chain complexes]] in $\mathcal{A}$. For \begin{displaymath} f : V^\bullet \to W^{\bullet} \end{displaymath} a morphism, the mapping cone is the complex \begin{displaymath} \begin{aligned} Cone(f) & \coloneqq (\cdots \to Cone(f)^{k-1} \stackrel{d_{Cone(f)}}{\to} Cone(f)^k) \to \cdots) \\ & \coloneqq \left( \itexarray{ \cdots \to & V^k &\stackrel{- d_V}{\to}& V^{k+1} & \to \cdots \\ & \oplus &\searrow^{f^k}& \oplus \\ \cdots \to & W^{k-1} &\underset{d_W}{\to}& W^k & \to \cdots } \right) \end{aligned} \,. \end{displaymath} There is a canonical co[[chain homotopy]] \begin{displaymath} \itexarray{ Cone(f) &\leftarrow& 0 \\ {}^{\mathllap{i}}\uparrow &\swArrow_{\eta}& \uparrow \\ W &\stackrel{f}{\leftarrow}& V } \end{displaymath} where $i : W \to Cone(f)$ is the canonical inclusion, componentwise given by \begin{displaymath} i^k : W^k \stackrel{(0,Id)}{\to} V_{k+1} \oplus W^k \end{displaymath} and where the cochain homotopy $\eta$ has components \begin{displaymath} \eta^k : V^k \stackrel{(Id,0)}{\to} Cone(f)^{k-1} = V^k \oplus W^{k-1} \end{displaymath} which we denote on $v \in V^k$ by \begin{displaymath} \eta : v \mapsto (f(v))[1] \,. \end{displaymath} The fact that this is a cochain homotopy means that \begin{displaymath} [d,\eta] = i \circ f - 0 \,, \end{displaymath} which we check on any $v \in V^k$ by computing \begin{displaymath} \begin{aligned} [d, \eta](v) &= d_{Cone(f)} \circ \eta (v) + \eta(d_V v) \\ & = d_{Cone(f)} (f(v)[1]) + (f(d_V v))[1] \\ & = \left( f(v) - (d_{W}f(v))[1] \right) + (d_W (f(v)))[1] \\ & = f(v) \end{aligned} \,, \end{displaymath} where we used the above definition of $d_{Cone(f)}$ and the fact that $f$ is a chain homomorphism and hence intertwines the differentials. This cochain homotopy is universal in that for any other cochain homotopy \begin{displaymath} \itexarray{ X &\leftarrow& 0 \\ {}^{\mathllap{j}}\uparrow &\swArrow_{\rho}& \uparrow \\ W &\stackrel{f}{\leftarrow}& V } \end{displaymath} hence \begin{displaymath} j \circ f = [d,\rho] \end{displaymath} we have a morphism \begin{displaymath} (j,\rho) : Cone(f) \to X \end{displaymath} given on $W$ by $j$ and on $V$ by $\rho$ \begin{displaymath} (j,\rho)^k : V^{k+1} \oplus W^k \stackrel{(\rho, j)}{\to} X^k \end{displaymath} which is indeed a cochain homomorphism because for all $v + w \in Cone(f)$ we have \begin{displaymath} \begin{aligned} d_X (j,\rho)(v + w) & = d_X (\rho v) + d_X (j(w)) \\ & = [d,\rho](v) - \rho d_V v + j (d_W w) \\ & = j f (v) - \rho d_V v + j (d_{Cone(f)} w) \\ & = (j,\rho) d_{Cone(f)}(v + w) \end{aligned} \end{displaymath} and which is unique with the property that [[whiskering]] of [[2-morphism]]s gives \begin{displaymath} \itexarray{ X &\leftarrow& 0 \\ {}^{\mathllap{j}}\uparrow &\swArrow_{\rho}& \uparrow \\ W &\stackrel{f}{\leftarrow}& V } \;\;\;\;\; = \;\;\;\;\; \itexarray{ X \\ & \nwarrow^{\mathrlap{(j,\rho)}} \\ && Cone(f) &\leftarrow& 0 \\ && {}^{\mathllap{i}}\uparrow &\swArrow_{\eta}& \uparrow \\ && W &\stackrel{f}{\leftarrow}& V } \end{displaymath} hence that \begin{displaymath} j = (j,\rho) \circ i \end{displaymath} and \begin{displaymath} \rho = (j,\rho) \circ \eta \,. \end{displaymath} \hypertarget{in_additive_categories_with_translation}{}\subsubsection*{{In additive categories with translation}}\label{in_additive_categories_with_translation} Let $\mathcal{A}$ be an [[additive category]] [[category with translation|with translation]] $T=[1] : \mathcal{A} \to \mathcal{A}$. Let $X$ and $Y$ be two [[differential objects]] in $(\mathcal{A},T)$ and $f : X \to Y$ any [[morphism]] in $C$. The \textbf{mapping cone} $Cone(f)$ of $f$ is the [[differential object]] whose underlying object is the [[direct sum]] $T X \oplus Y$ and whose differential $d_{cone f} : T X \oplus X \to T T X \oplus T X$ is given in [[matrix calculus]] notation by \begin{displaymath} d_{cone f} := \left( \itexarray{ d_{T X} & 0 \\ T(f) & d_Y } \right) = \left( \itexarray{ - T(d_X) & 0 \\ T(f) & d_Y } \right) \,. \end{displaymath} Notice the minus sign here, coming from the definition of a [[differential object|shifted differential object]]. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{HomologyExactSequencesAndFiberSequences}{}\subsubsection*{{Homology exact sequences and fiber sequences}}\label{HomologyExactSequencesAndFiberSequences} We discuss the relation between mapping cones in [[categories of chain complexes]], as \hyperlink{InChainComplexes}{above}, and [[long exact sequences in homology]]. For an exposition of the following see there the section \emph{\href{long+exact+sequence+in+homology#RelationToHomotopyFiberSequences}{Relation to homotopy fiber sequences}}. Let $f : X_\bullet \longrightarrow Y_\bullet$ be a [[chain map]] and write $cone(f) \in Ch_\bullet(\mathcal{A})$ for its mapping cone as explicitly given in prop. \ref{ComponentsOfMappingConeInChainComplexes}. \begin{defn} \label{ProjectionOutOfChainComplexMappingCone}\hypertarget{ProjectionOutOfChainComplexMappingCone}{} Write $X[1]_\bullet \in Ch_\bullet(\mathcal{A})$ for the [[suspension of a chain complex]] of $X$. Write \begin{displaymath} p : cone(f) \to X[1]_\bullet \end{displaymath} for the [[chain map]] which in components \begin{displaymath} p_n : cone(f)_n \to X[1]_n \end{displaymath} is given, via prop. \ref{ComponentsOfMappingConeInChainComplexes}, by the canonical projection out of a direct sum \begin{displaymath} p_n : Y_\n \oplus X_{n-1} \to X_{n-1} \,. \end{displaymath} \end{defn} \begin{prop} \label{ProjectionOutOfChainComplexMappingConeIsHoCofiber}\hypertarget{ProjectionOutOfChainComplexMappingConeIsHoCofiber}{} The chain map $p : cone(f)_\bullet \to X[1]_\bullet$ represents the [[homotopy cofiber]] of the canonical map $i : Y_\bullet \to cone(f)_\bullet$. \end{prop} \begin{proof} By prop. \ref{InclusionIntoChainComplexCone} and def. \ref{ProjectionOutOfChainComplexMappingCone} the sequence \begin{displaymath} Y_\bullet \stackrel{i}{\to} cone(f)_\bullet \stackrel{p}{\to} X[1]_\bullet \end{displaymath} is a [[short exact sequence]] of chain complexes (since it is so degreewise, in fact degreewise it is even a [[split exact sequence]]). In particular we have a [[cofiber]] [[pushout]] diagram \begin{displaymath} \itexarray{ Y_\bullet &\stackrel{i}{\hookrightarrow}& cone(f)_\bullet \\ \downarrow && \downarrow \\ 0 &\to& X[1]_\bullet } \,. \end{displaymath} Now, in the [[injective model structure on chain complexes]] all chain complxes are [[cofibrant objects]] and an inclusion such as $i : Y_\bullet \hookrightarrow cone(f)_\bullet$ is a [[cofibration]]. By the detailed discussion at [[homotopy limit]] this means that the ordinary colimit here is in fact a [[homotopy colimit]], hence exhibits $p$ as the [[homotopy cofiber]] of $i$. \end{proof} \begin{cor} \label{HomotopyCofiberSequenceOfChainMap}\hypertarget{HomotopyCofiberSequenceOfChainMap}{} For $f_\bullet : X_\bullet \to Y_\bullet$ a [[chain map]], there is a \textbf{[[homotopy cofiber sequence]]} of the form \begin{displaymath} X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{i_\bullet}{\to} cone(f)_\bullet \stackrel{p_\bullet}{\to} X[1]_\bullet \stackrel{f[1]_\bullet}{\to} Y_\bullet \stackrel{i[1]_\bullet}{\to} cone(f)_\bullet \stackrel{p[1]_\bullet}{\to} X[2]_\bullet \to \cdots \end{displaymath} \end{cor} In order to compare this to the discussion of [[nLab:connecting homomorphisms]], we now turn attention to the case that $f_\bullet$ happens to be a [[monomorphism]]. Notice that this we can always assume, up to [[quasi-isomorphism]], for instance by prolonging $f$ by the map into its [[mapping cylinder]] \begin{displaymath} X_\bullet \to Y_\bullet \stackrel{\simeq}{\to} cyl(f) \,. \end{displaymath} By the axioms on an [[abelian category]] in this case we have a [[short exact sequence]] \begin{displaymath} 0 \to X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet \to 0 \end{displaymath} of chain complexes. The following discussion revolves around the fact that now $cone(f)_\bullet$ as well as $Z_\bullet$ are both models for the homotopy cofiber of $f$. \begin{lemma} \label{ConeInjectionEquivalentToZigzag}\hypertarget{ConeInjectionEquivalentToZigzag}{} Let \begin{displaymath} X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet \end{displaymath} be a [[short exact sequence]] of [[chain complexes]]. The collection of linear maps \begin{displaymath} h_n : Y_n \oplus X_{n-1} \to Y_n \stackrel{}{\to} Z_n \end{displaymath} constitutes a [[chain map]] \begin{displaymath} h_\bullet : cone(f)_\bullet \to Z_\bullet \,. \end{displaymath} This is a [[quasi-isomorphism]]. The inverse of $H_n(h_\bullet)$ is given by sending a representing [[cycle]] $z \in Z_n$ to \begin{displaymath} (\hat z_n, \partial^Y \hat z_n) \in Y_n \oplus X_{n+1} \,, \end{displaymath} where $\hat z_n$ is any choice of lift through $p_n$ and where $\partial^Y \hat z_n$ is the formula expressing the [[connecting homomorphism]] in terms of elements, as discussed at \href{connecting%20homomorphism#OnHomologyInTermsOfElements}{Connecting homomorphism -- In terms of elements}. Finally, the morphism $i_\bullet : Y_\bullet \to cone(f)_\bullet$ is eqivalent in the [[homotopy category]] (the [[derived category]]) to the [[zigzag]] \begin{displaymath} \itexarray{ && cone(f)_\bullet \\ && \downarrow^{\mathrlap{h}}_{\mathrlap{\simeq}} \\ Y_\bullet &\to& Z_\bullet } \,. \end{displaymath} \end{lemma} In the literature this appears for instance as (\hyperlink{Schapira}{Schapira, cor. 7.2.2}). \begin{proof} To see that $h_\bullet$ defines a chain map recall the differential $\partial^{cone(f)}$ from prop. \ref{ComponentsOfMappingConeInChainComplexes}, which acts by \begin{displaymath} \partial^{cone(f)} (x_{n-1}, \hat z_n) = ( -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} ) \end{displaymath} and use that $x_{n-1}$ is in the [[kernel]] of $p_n$ by exactness, hence \begin{displaymath} \begin{aligned} h_{n-1}\partial^{cone(f)}(x_{n-1}, \hat z_n) &= h_{n-1}( -\partial^X x_{n-1}, \partial^Y \hat z_n + x_{n-1} ) \\ & = p_{n-1}( \partial^Y \hat z_n + x_{n-1}) \\ & = p_{n-1}( \partial^Y \hat z_n ) \\ & = \partial^Z p_n \hat z_n \\ & = \partial^Z h_n(x_{n-1}, \hat z_n) \end{aligned} \,. \end{displaymath} It is immediate to see that we have a [[commuting diagram]] of the form \begin{displaymath} \itexarray{ && cone(f)_\bullet \\ & {}^{\mathllap{i_\bullet}}\nearrow& \downarrow^{\mathrlap{h}}_{\mathrlap{\simeq}} \\ Y_\bullet &\to& Z_\bullet } \end{displaymath} since the composite morphism is the inclusion of $Y$ followed by the bottom morphism on $Y$. Abstractly, this already implies that $cone(f)_\bullet \to Z_\bullet$ is a [[quasi-isomorphism]], for this diagram gives a morphism of [[cocones]] under the diagram defining $cone(f)$ in prop. \ref{HomotopyCofiberByFactorizationLemma} and by the above both of these cocones are [[homotopy colimit|homotopy-colimiting]]. But in checking the claimed inverse of the induced map on homology groups, we verify this also explicity: We first determine those cycles $(x_{n-1}, y_n) \in cone(f)_n$ which lift a cycle $z_n$. By lemma \ref{HomotopyCofiberByFactorizationLemma} a lift of chains is any pair of the form $(x_{n-1}, \hat z_n)$ where $\hat z_n$ is a lift of $z_n$ through $Y_n \to X_n$. So $x_{n-1}$ has to be found such that this pair is a cycle. By prop. \ref{ComponentsOfMappingConeInChainComplexes} the differential acts on it by \begin{displaymath} \partial^{cone(f)} (x_{n-1}, \hat z_n) = ( -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} ) \end{displaymath} and so the condition is that $x_{n-1} \coloneqq -\partial^Y \hat z_n$ (which implies $\partial^X x_{n-1} = -\partial^X \partial^Y \hat z_n = -\partial^Y \partial^Y \hat z_n = 0$ due to the fact that $f_n$ is assumed to be an inclusion, hence that $\partial^X$ is the restriction of $\partial^Y$ to elements in $X_n$). This condition clearly has a unique solution for every lift $\hat z_n$ and a lift $\hat z_n$ always exists since $p_n : Y_n \to Z_n$ is surjective, by assumption that we have a [[short exact sequence]] of chain complexes. This shows that $H_n(h_\bullet)$ is surjective. To see that it is also injective we need to show that if a [[cycle]] $(-\partial^Y \hat z_n, \hat z_n) \in cone(f)_n$ maps to a cycle $z_n = p_n(\hat z_n)$ that is trivial in $H_n(Z)$ in that there is $c_{n+1}$ with $\partial^Z c_{n+1} = z_n$, then also the original cycle was trivial in homology, in that there is $(x_n, y_{n+1})$ with \begin{displaymath} \partial^{cone(f)}(x_n, y_{n+1}) \coloneqq (-\partial^X x_n, \partial^Y y_{n+1} + x_n) = (-\partial^Y \hat z_n, \hat z_n) \,. \end{displaymath} For that let $\hat c_{n+1} \in Y_{n+1}$ be a lift of $c_{n+1}$ through $p_n$, which exists again by surjectivity of $p_{n+1}$. Observe that \begin{displaymath} p_{n}( \hat z_n - \partial^Y \hat c_{n+1}) = z_n -\partial^Z ( p_n \hat c_{n+1} ) = z_n - \partial^Z ( c_{n+1} ) = 0 \end{displaymath} by assumption on $z_n$ and $c_{n+1}$, and hence that $\hat z_n - \partial^Y \hat c_{n+1}$ is in $X_n$ by exactness. Hence $(z_n - \partial^Y \hat c_{n+1}, \hat c_{n+1}) \in cone(f)_n$ trivializes the given cocycle: \begin{displaymath} \begin{aligned} \partial^{cone(f)}( \hat z_n - \partial^Y \hat c_{n+1} , \hat c_{n+1}) & = (-\partial^X(\hat z_n - \partial^Y \hat c_{n+1} ), \partial^Y \hat c_{n+1} + (\hat z_n - \partial^Y \hat c_{n+1} ) ) \\ & = (-\partial^Y(\hat z_n - \partial^Y \hat c_{n+1}), \hat z_n ) \\ & = ( -\partial^Y \hat z_n, \hat z_n ) \end{aligned} \,. \end{displaymath} \end{proof} \begin{theorem} \label{}\hypertarget{}{} Let \begin{displaymath} X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \to Z_\bullet \end{displaymath} be a [[short exact sequence]] of [[chain complexes]]. Then the [[chain homology]] functor \begin{displaymath} H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A} \end{displaymath} sends the homotopy cofiber sequence of $f$, cor. \ref{HomotopyCofiberSequenceOfChainMap}, to the [[long exact sequence in homology]] induced by the given short exact sequence, hence to \begin{displaymath} H_n(X_\bullet) \to H_n(Y_\bullet) \to H_n(Z_\bullet) \stackrel{\delta}{\to} H_{n-1}(X_\bullet) \to H_{n-1}(Y_\bullet) \to H_{n-1}(Z_\bullet) \stackrel{\delta}{\to} H_{n-2}(X_\bullet) \to \cdots \,, \end{displaymath} where $\delta_n$ is the $n$th [[connecting homomorphism]]. \end{theorem} \begin{proof} By lemma \ref{ConeInjectionEquivalentToZigzag} the homotopy cofiber sequence is equivalen to the [[zigzag]] \begin{displaymath} \itexarray{ && && && && && cone(f)[1]_\bullet &\to& \cdots \\ && && && && && \downarrow^{\mathrlap{h[1]_\bullet}}_{\mathrlap{\simeq}} \\ && && cone(f)_\bullet &\to& X[1]_\bullet &\stackrel{f[1]_\bullet}{\to}& Y[1]_\bullet &\to& Z[1]_\bullet \\ && && \downarrow^{\mathrlap{h_\bullet}}_{\mathrlap{\simeq}} \\ X_\bullet &\stackrel{f_\bullet}{\to}& Y_\bullet &\stackrel{}{\to}& Z_\bullet } \,. \end{displaymath} Observe that \begin{displaymath} H_n( X[k]_\bullet) \simeq H_{n-k}(X_\bullet) \,. \end{displaymath} It is therefore sufficient to check that \begin{displaymath} H_n \left( \itexarray{ cone(f)_\bullet &\to& X[1]_\bullet \\ \downarrow^{\mathrlap{\simeq}} \\ Z_\bullet } \right) \;\; : \;\; H_n(Z_\bullet) \to H_n(cone(f)_\bullet) \to H_{n-1}(X_\bullet) \end{displaymath} equals the [[connecting homomorphism]] $\delta_n$ induced by the short exact sequence. By prop. \ref{ConeInjectionEquivalentToZigzag} the inverse of the vertical map is given by choosing lifts and forming the corresponding element given by the connecting homomorphism. By prop. \ref{ProjectionOutOfChainComplexMappingConeIsHoCofiber} the horizontal map is just the projection, and hence the assignment is of the form \begin{displaymath} [z_n] \mapsto [x_{n-1}, y_n] \mapsto [x_{n-1}] \,. \end{displaymath} So in total the image of the zig-zag under homology sends \begin{displaymath} [z_n]_Z \mapsto -[\partial^Y \hat z_n]_X \,. \end{displaymath} By the discussion \href{connecting%20homomorphism#OnHomologyInTermsOfElements}{there}, this is indeed the action of the [[connecting homomorphism]]. \end{proof} \hypertarget{DistinguishedTriangles}{}\subsubsection*{{Distinguished triangles from mapping cones}}\label{DistinguishedTriangles} In summary, the \hyperlink{HomologyExactSequencesAndFiberSequences}{above} says that for every [[chain map]] $f_\bullet : X_\bullet \to Y_\bullet$ we obtain maps \begin{displaymath} X_\bullet \stackrel{f}{\to} Y_\bullet \stackrel{ \left( \itexarray{ 0 \\ id_{Y_\bullet} } \right) }{\to} cone(f)_\bullet \stackrel{ \left( \itexarray{ id_{X[1]_\bullet} & 0 } \right) }{\to} X[1]_\bullet \end{displaymath} which form a [[homotopy fiber sequence]] and such that this sequence continues by forming [[suspension of a chain complex|suspensions]], hence for all $n \in \mathbb{Z}$ we have \begin{displaymath} X[n]_\bullet \stackrel{f}{\to} Y[n]_\bullet \stackrel{ \left( \itexarray{ 0 \\ id_{Y[n]_\bullet} } \right) }{\to} cone(f)[n]_\bullet \stackrel{ \left( \itexarray{ id_{X[n+11]_\bullet} & 0 } \right) }{\to} X[n+1]_\bullet \end{displaymath} To amplify this quasi-cyclic behaviour one sometimes depicts the situation as follows: \begin{displaymath} \itexarray{ X_\bullet &&\stackrel{f}{\to}&& Y_\bullet \\ & {}_{\mathllap{[1]}}\nwarrow && \swarrow \\ && cone(f)_\bullet } \end{displaymath} and hence speaks of a ``triangle'', or \textbf{distinguished triangle} or \textbf{mapping cone triangle} of $f$. \begin{itemize}% \item distinguished triangle = period of [[homotopy fiber sequence]] . \end{itemize} Due to these ``triangles'' one calls the [[homotopy category]] of chain complexes [[localization|localized]] at the [[quasi-isomorphisms]], hence the [[derived category]], a \textbf{[[triangulated category]]}. Notice that equivalently we can express the triangles via the [[mapping cylinder]]. For every map of [[chain complex]]es $f:A\to B$, the cylinder $Cyl(f)$ is quasi-isomorphic to $B$, and moreover in the [[homotopy category]] of chain complexes, every distinguished triangle is quasi-isomorphic to a distinguished triangle of the form \begin{displaymath} A\to Cyl(u)\to Cone(u)\to A[1] \end{displaymath} for some $u:A\to B$ where all the morphisms in the triangle are appropriatedly induced by $u$. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[!include universal constructions of topological spaces -- table]] \hypertarget{references}{}\subsection*{{References}}\label{references} In the context of [[chain complexes]] the construction is discussed for instance in \begin{itemize}% \item [[Pierre Schapira]], section 3.2 and section 7 of \emph{Categories and homological algebra} (2011) (\href{http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf}{pdf}) \item [[Charles Weibel]], section 1.5 of \emph{[[An Introduction to Homological Algebra]]} . \end{itemize} In the context of [[spectra]] discussion includes \begin{itemize}% \item [[Robert Switzer]], around 8. 17 of \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975. \end{itemize} [[!redirects mapping cones]] \end{document}