nLab mapping cone

Idea

The 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 cofiber of $f$.

It is also called the homotopy cokernel of $f$ or the weak quotient of $Y$ by the image of $X$ in $Y$ under $f$.

The dual notion is that of mapping cocone.

(graphics taken from Muro 2010)

Definition

The mapping cone construction is a means to present in a category with weak equivalences the following canonical construction in homotopy theory/(∞,1)-category theory.

Definition

In an (∞,1)-category $\mathcal{C}$ with terminal object and (∞,1)-pushout, the homotopy cofiber of a morphism $f : X \to Y$ is the homotopy pushout

$$coker(f) \coloneqq Y \coprod_X {*}$$

hence the object universal construction sitting universally in a diagram of the form

$$\array{ X &\stackrel{}{\to}& {*} \\ \downarrow^{\mathrlap{f}} &\swArrow_{\simeq}& \downarrow \\ Y &\to& coker(f) } \,.$$

Proposition

If the (∞,1)-category $\mathcal{C}$ is presented by (is 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

$$\array{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\to}& cyl(X) \\ \downarrow && &\searrow & \downarrow \\ {*} &\to& &\to& cone(f) }$$

in $C$ using any cylinder object $cyl(X)$ for $X$.

This is discussed in detail at factorization lemma and at homotopy pullback.

Remark

Intuitively this says that $cone(f)$ is the object obtained by

1. forming the cylinder over $X$;

2. gluing to one end of that the object $Y$ as specified by the map $f$.

3. shrinking the other end of the cylinder to the point.

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 cone 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”.

Remark

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

$$\array{ X \\ \downarrow^{\mathrlap{i_0}} & \searrow^{\mathrlap{g}} \\ cyl(X) &\stackrel{\eta}{\to}& Y \\ \uparrow^{\mathrlap{i_1}} & \nearrow_{\mathrlap{h}} \\ X } \,.$$

Therefore prop. says that the mapping cone is the the 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. is saying.

Proposition

The colimit in prop. may be computed in two stages by two consecutive pushouts in $C$, and in two ways by the following pasting diagram:

$$\array{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{i_1} && \downarrow \\ X &\stackrel{i_0}{\to}& cyl(X) &\to & cyl(f) \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& cone(X) &\to& cone(f) } \,.$$

Here every square is a pushout, (and so by the pasting law is every rectangular pasting composite).

This now is a basic fact in ordinary category theory. The pushouts appearing here go by the following names:

Definition

The pushout

$$\array{ X &\stackrel{i_0}{\to}& cyl(X) \\ \downarrow && \downarrow \\ {*} &\to& cone(X) }$$

defines the cone $cone(X)$ over $X$ (with respect to the chosen cylinder object): the result of taking the cylinder over $X$ and identifying one $X$-shaped end with the point.

The pushout

$$\array{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow \\ cyl(X) &\to& cyl(f) }$$

defines the 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

$$\array{ cyl(X) &\to& cyl(f) \\ \downarrow && \downarrow \\ cone(X) &\to& cone(f) }$$

defines the 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 $f$.

The geometric intuition behind this is best seen in the archetypical example of the classical model structure on topological spaces. See the example For topological spaces below. The example For chain complexes can be understood similarly geometrically by thinking of all chain complexes as singular chains on topological spaces.

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 for topological spaces and that for chain complexes.

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 $X$.

For topological spaces

The notion mapping cone derives its name from its geometrical interpretation in the category Top of topological spaces.

For more details see also at 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 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

$$cyl(X) \coloneqq X \times I \,,$$

which is literally the cylinder over $X$.

Given a continuous function $f:X\to Y$, the topological space $cone(f)$ is

$$cone(f) = (X \times I) \cup_{f} Y$$

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).

For chain complexes

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 . The end result is prop. below, reproducing the classical formula for the mapping cone.

Definition

Write $*_\bullet \in Ch_\bullet(\mathcal{A})$ for the chain complex concentrated on $R$ in degree 0

$$*_\bullet 0 = [\cdots \to 0 \to 0 \to R] \,.$$

Remark

This may be understood as the normalized chain complex of chains of simplices on the terminal simplicial set $\Delta^0$, the 0-simplex.

Definition

Let $I_\bullet \in Ch_{\bullet}(\mathcal{A})$ be given by

$$I_\bullet = (\cdots 0 \to 0 \to R \stackrel{(-id,id)}{\to} R \oplus R) \,.$$

Denote by

$$i_0 : *_\bullet \to I_\bullet$$

the chain map which in degree 0 is the canonical inclusion into the second summand of a direct sum and by

$$i_1 : *_\bullet \to I_\bullet$$

correspondingly the canonical inclusion into the first summand.

Remark

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:

$$I_\bullet = C_\bullet(\Delta[1]) \,.$$

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

$$\partial ( 0 \to 1 ) = (1) - (0) \,.$$

We decompose the proof of this statement is a sequence of substatements.

Proposition

For $X_\bullet \in Ch_\bullet$ the tensor product of chain complexes

$$(I \otimes X)_\bullet \in Ch_\bullet$$

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).

Proposition

The complex $(I \otimes X)_\bullet$ has components

$$(I \otimes X)_n = X_n \oplus X_n \oplus X_{n-1}$$

and the differential is given by

$$\array{ X_{n+1} \oplus X_{n+1} &\stackrel{\partial^X \oplus \partial^X}{\to}& X_n \oplus X_n \\ \oplus &\nearrow_{(-id,id)}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,,$$

hence in matrix calculus by

$$\partial^{I \otimes X} = \left( \array{ \partial^X \oplus \partial^X & (-id, id) \\ 0 & -\partial^X } \right) \;\;\colon\;\; (X_{n+1} \oplus X_{n+1}) \oplus X_{n} \to (X_{n} \oplus X_{n}) \oplus X_{n-1} \,.$$

Proof

By the formula discussed at tensor product of chain complexes the components arise as the direct sum

$$(I \otimes X )_n = (R_{(0)} \otimes X_n ) \oplus (R_{(1)} \otimes X_n ) \oplus (R_{(0 \to 1)} \otimes X_{(n-1)} )$$

and the differential picks up a sign when passed past the degree-1 term $R_{(0 \to 1)}$:

$$\begin{aligned} \partial^{I \otimes X} ( (0 \to 1), x ) &= ( (\partial^I (0 \to 1)), x ) - ( (0\to 1), \partial^X x ) \\ & = ( - (0) + (1), x ) - ( (0 \to 1), \partial^X x ) \\ & = -((0), x) + ((1), x) - ( (0 \to 1), \partial^X x ) \end{aligned} \,.$$

Remark

The two boundary inclusions of $X_\bullet$ into the cylinder are given in terms of def. by

$$i^X_0 : X_\bullet \simeq *_\bullet \otimes X_\bullet \stackrel{i_0 \otimes id_X}{\to} (I\otimes X)_\bullet$$

and

$$i^X_1 : X_\bullet \simeq *_\bullet \otimes X_\bullet \stackrel{i_1 \otimes id_X}{\to} (I\otimes X)_\bullet$$

which in components is the inclusion of the second or first direct summand, respectively

$$X_n \hookrightarrow X_n \oplus X_n \oplus X_{n-1} \,.$$

One part of definition now reads:

Definition

For $f_\bullet : X_\bullet \to Y_\bullet$ a chain map, the mapping cylinder $cyl(f)$ is the pushout

$$\array{ cyl(f)_\bullet &\leftarrow& Y_\bullet \\ \uparrow && \uparrow^{\mathrlap{f}} \\ I_\bullet \otimes X_\bullet &\stackrel{i_0}{\leftarrow}& X_\bullet } \,.$$

Proposition

The components of $cyl(f)_\bullet$ are

$$cyl(f)_n = X_n \oplus Y_n \oplus X_{n-1}$$

and the differential is given by

$$\array{ X_{n+1} \oplus Y_{n+1} &\stackrel{\partial^X \oplus \partial^Y}{\to}& X_n \oplus Y_n \\ \oplus &\nearrow_{(-id,f)}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,,$$

hence in matrix calculus by

$$\partial^{cyl(f)} = \left( \array{ \partial^X \oplus \partial^Y & (-id, f_n) \\ 0 & -\partial^X } \right) : (X_{n+1} \oplus Y_{n+1}) \oplus X_{n} \to (X_{n} \oplus Y_{n}) \oplus X_{n-1} \,.$$

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 limits in presheaf categories). Here the statement is evident:

the pushout identifies one 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 $f_n$.

The last part of definition now reads:

Definition

For $f_\bullet : X_\bullet \to Y_\bullet$ a chain map, the mapping cone $cone(f)$ is the pushout

$$\array{ cone(f) &\leftarrow& cyl(f) \\ \uparrow && \uparrow \\ cone(X) &\leftarrow& X \otimes I \\ \uparrow && \uparrow^{\mathrlap{i_1}} \\ 0 &\leftarrow& X }$$

In the literature this appears for instance as (Schapira, def. 3.2.2).

Proposition

The components of the mapping cone $cone(f)$ are

$$cone(f)_n = Y_n \oplus X_{n-1}$$

with differential given by

$$\array{ Y_{n+1} &\stackrel{\partial^Y}{\to}& Y_n \\ \oplus &\nearrow_{f_n}& \oplus \\ X_{n} &\underset{-\partial^X}{\to}& X_{n-1} } \,,$$

and hence in matrix calculus by

$$\partial^{cone(f)} = \left( \array{ \partial^Y_{n+1} & f_n \\ 0 & -\partial^X_n } \right) : Y_{n+1} \oplus X_{n} \to Y_{n} \oplus X_{n-1} \,.$$

Proof

As before the pushout is computed degreewise. This identifies the remaining unshifted copy of $X$ with 0.

Proposition

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

$$i_n : Y_n \to cone(f)_n = Y_n \oplus X_{n-1}$$

by the canonical inclusion of a summand into a direct sum.

Proof

This follows by starting with remark and then following these inclusions through the formation of the two colimits as discussed above.

The construction above builds the mapping cone explicitly via the standard formula for homotopy pushouts. Often however other presentations are more convenient:

Proposition

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 $f$:

$$Cone(f) \simeq tot(D_{\bullet,\bullet}) \,.$$

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).

This appears for instance as (Weibel, Exercise 1.2.8).

For cochain complexes

We spell out the situation in more detail in a category of cochain complexes.

Let $\mathcal{A}$ be some concrete additive category and $Ch^\bullet(\mathcal{A})$ the category of chain complexes in $\mathcal{A}$. For

$$f : V^\bullet \to W^{\bullet}$$

a morphism, the mapping cone is the complex

$$\begin{aligned} Cone(f) & \coloneqq (\cdots \to Cone(f)^{k-1} \stackrel{d_{Cone(f)}}{\to} Cone(f)^k) \to \cdots) \\ & \coloneqq \left( \array{ \cdots \to & V^k &\stackrel{- d_V}{\to}& V^{k+1} & \to \cdots \\ & \oplus &\searrow^{f^k}& \oplus \\ \cdots \to & W^{k-1} &\underset{d_W}{\to}& W^k & \to \cdots } \right) \end{aligned} \,.$$

There is a canonical cochain homotopy

$$\array{ Cone(f) &\leftarrow& 0 \\ {}^{\mathllap{i}}\uparrow &\swArrow_{\eta}& \uparrow \\ W &\stackrel{f}{\leftarrow}& V }$$

where $i : W \to Cone(f)$ is the canonical inclusion, componentwise given by

$$i^k : W^k \stackrel{(0,Id)}{\to} V_{k+1} \oplus W^k$$

and where the cochain homotopy $\eta$ has components

$$\eta^k : V^k \stackrel{(Id,0)}{\to} Cone(f)^{k-1} = V^k \oplus W^{k-1}$$

which we denote on $v \in V^k$ by

$$\eta : v \mapsto (f(v))[1] \,.$$

The fact that this is a cochain homotopy means that

$$[d,\eta] = i \circ f - 0 \,,$$

which we check on any $v \in V^k$ by computing

$$\begin{aligned} [d, \eta](v) &= d_{Cone(f)} \circ \eta (v) + \eta(d_V v) \\ & = d_{Cone(f)} (f(v)[1]) + (f(d_V v))[1] \\ & = \left( f(v) - (d_{W}f(v))[1] \right) + (d_W (f(v)))[1] \\ & = f(v) \end{aligned} \,,$$

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

$$\array{ X &\leftarrow& 0 \\ {}^{\mathllap{j}}\uparrow &\swArrow_{\rho}& \uparrow \\ W &\stackrel{f}{\leftarrow}& V }$$

hence

$$j \circ f = [d,\rho]$$

we have a morphism

$$(j,\rho) : Cone(f) \to X$$

given on $W$ by $j$ and on $V$ by $\rho$

$$(j,\rho)^k : V^{k+1} \oplus W^k \stackrel{(\rho, j)}{\to} X^k$$

which is indeed a cochain homomorphism because for all $v + w \in Cone(f)$ we have

$$\begin{aligned} d_X (j,\rho)(v + w) & = d_X (\rho v) + d_X (j(w)) \\ & = [d,\rho](v) - \rho d_V v + j (d_W w) \\ & = j f (v) - \rho d_V v + j (d_{Cone(f)} w) \\ & = (j,\rho) d_{Cone(f)}(v + w) \end{aligned}$$

and which is unique with the property that whiskering of 2-morphisms gives

$$\array{ X &\leftarrow& 0 \\ {}^{\mathllap{j}}\uparrow &\swArrow_{\rho}& \uparrow \\ W &\stackrel{f}{\leftarrow}& V } \;\;\;\;\; = \;\;\;\;\; \array{ X \\ & \nwarrow^{\mathrlap{(j,\rho)}} \\ && Cone(f) &\leftarrow& 0 \\ && {}^{\mathllap{i}}\uparrow &\swArrow_{\eta}& \uparrow \\ && W &\stackrel{f}{\leftarrow}& V }$$

hence that

$$j = (j,\rho) \circ i$$

and

$$\rho = (j,\rho) \circ \eta \,.$$

In additive categories with translation

Let $\mathcal{A}$ be an additive category 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 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

$$d_{cone f} := \left( \array{ d_{T X} & 0 \\ T(f) & d_Y } \right) = \left( \array{ - T(d_X) & 0 \\ T(f) & d_Y } \right) \,.$$

Notice the minus sign here, coming from the definition of a shifted differential object.

Properties

Homology exact sequences and fiber sequences

We discuss the relation between mapping cones in categories of chain complexes, as above, and long exact sequences in homology. For an exposition of the following see there the section 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. .

Definition

Write $X[1]_\bullet \in Ch_\bullet(\mathcal{A})$ for the suspension of a chain complex of $X$. Write

$$p : cone(f) \to X[1]_\bullet$$

for the chain map which in components

$$p_n : cone(f)_n \to X[1]_n$$

is given, via prop. , by the canonical projection out of a direct sum

$$p_n : Y_\n \oplus X_{n-1} \to X_{n-1} \,.$$

Proposition

The chain map $p : cone(f)_\bullet \to X[1]_\bullet$ represents the homotopy cofiber of the canonical map $i : Y_\bullet \to cone(f)_\bullet$.

Proof

By prop. and def. the sequence

$$Y_\bullet \stackrel{i}{\to} cone(f)_\bullet \stackrel{p}{\to} X[1]_\bullet$$

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

$$\array{ Y_\bullet &\stackrel{i}{\hookrightarrow}& cone(f)_\bullet \\ \downarrow && \downarrow \\ 0 &\to& X[1]_\bullet } \,.$$

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 $i$.

Corollary

For $f_\bullet : X_\bullet \to Y_\bullet$ a chain map, there is a homotopy cofiber sequence of the form

$$X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{i_\bullet}{\to} cone(f)_\bullet \stackrel{p_\bullet}{\to} X[1]_\bullet \stackrel{f[1]_\bullet}{\to} Y_\bullet \stackrel{i[1]_\bullet}{\to} cone(f)_\bullet \stackrel{p[1]_\bullet}{\to} X[2]_\bullet \to \cdots$$

In order to compare this to the discussion of 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

$$X_\bullet \to Y_\bullet \stackrel{\simeq}{\to} cyl(f) \,.$$

By the axioms on an abelian category in this case we have a short exact sequence

$$0 \to X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet \to 0$$

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 $f$.

Lemma

Let

$$X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \stackrel{p_\bullet}{\to} Z_\bullet$$

be a short exact sequence of chain complexes.

The collection of linear maps

$$h_n : Y_n \oplus X_{n-1} \to Y_n \stackrel{}{\to} Z_n$$

constitutes a chain map

$$h_\bullet : cone(f)_\bullet \to Z_\bullet \,.$$

This is a quasi-isomorphism. The inverse of $H_n(h_\bullet)$ is given by sending a representing cycle $z \in Z_n$ to

$$(\hat z_n, \partial^Y \hat z_n) \in Y_n \oplus X_{n+1} \,,$$

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 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

$$\array{ && cone(f)_\bullet \\ && \downarrow^{\mathrlap{h}}_{\mathrlap{\simeq}} \\ Y_\bullet &\to& Z_\bullet } \,.$$

In the literature this appears for instance as (Schapira, cor. 7.2.2).

Proof

To see that $h_\bullet$ defines a chain map recall the differential $\partial^{cone(f)}$ from prop. , which acts by

$$\partial^{cone(f)} (x_{n-1}, \hat z_n) = ( -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} )$$

and use that $x_{n-1}$ is in the kernel of $p_n$ by exactness, hence

$$\begin{aligned} h_{n-1}\partial^{cone(f)}(x_{n-1}, \hat z_n) &= h_{n-1}( -\partial^X x_{n-1}, \partial^Y \hat z_n + x_{n-1} ) \\ & = p_{n-1}( \partial^Y \hat z_n + x_{n-1}) \\ & = p_{n-1}( \partial^Y \hat z_n ) \\ & = \partial^Z p_n \hat z_n \\ & = \partial^Z h_n(x_{n-1}, \hat z_n) \end{aligned} \,.$$

It is immediate to see that we have a commuting diagram of the form

$$\array{ && cone(f)_\bullet \\ & {}^{\mathllap{i_\bullet}}\nearrow& \downarrow^{\mathrlap{h}}_{\mathrlap{\simeq}} \\ Y_\bullet &\to& Z_\bullet }$$

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. and by the above both of these cocones are 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 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. the differential acts on it by

$$\partial^{cone(f)} (x_{n-1}, \hat z_n) = ( -\partial^X x_{n-1} , \partial^Y \hat z_n + x_{n-1} )$$

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

$$\partial^{cone(f)}(x_n, y_{n+1}) \coloneqq (-\partial^X x_n, \partial^Y y_{n+1} + x_n) = (-\partial^Y \hat z_n, \hat z_n) \,.$$

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

$$p_{n}( \hat z_n - \partial^Y \hat c_{n+1}) = z_n -\partial^Z ( p_n \hat c_{n+1} ) = z_n - \partial^Z ( c_{n+1} ) = 0$$

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:

$$\begin{aligned} \partial^{cone(f)}( \hat z_n - \partial^Y \hat c_{n+1} , \hat c_{n+1}) & = (-\partial^X(\hat z_n - \partial^Y \hat c_{n+1} ), \partial^Y \hat c_{n+1} + (\hat z_n - \partial^Y \hat c_{n+1} ) ) \\ & = (-\partial^Y(\hat z_n - \partial^Y \hat c_{n+1}), \hat z_n ) \\ & = ( -\partial^Y \hat z_n, \hat z_n ) \end{aligned} \,.$$

Theorem

Let

$$X_\bullet \stackrel{f_\bullet}{\to} Y_\bullet \to Z_\bullet$$

be a short exact sequence of chain complexes.

Then the chain homology functor

$$H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}$$

sends the homotopy cofiber sequence of $f$, cor. , to the long exact sequence in homology induced by the given short exact sequence, hence to

$$H_n(X_\bullet) \to H_n(Y_\bullet) \to H_n(Z_\bullet) \stackrel{\delta}{\to} H_{n-1}(X_\bullet) \to H_{n-1}(Y_\bullet) \to H_{n-1}(Z_\bullet) \stackrel{\delta}{\to} H_{n-2}(X_\bullet) \to \cdots \,,$$

where $\delta_n$ is the $n$th connecting homomorphism.

Proof

By lemma the homotopy cofiber sequence is equivalen to the zigzag

$$\array{ && && && && && cone(f)[1]_\bullet &\to& \cdots \\ && && && && && \downarrow^{\mathrlap{h[1]_\bullet}}_{\mathrlap{\simeq}} \\ && && cone(f)_\bullet &\to& X[1]_\bullet &\stackrel{f[1]_\bullet}{\to}& Y[1]_\bullet &\to& Z[1]_\bullet \\ && && \downarrow^{\mathrlap{h_\bullet}}_{\mathrlap{\simeq}} \\ X_\bullet &\stackrel{f_\bullet}{\to}& Y_\bullet &\stackrel{}{\to}& Z_\bullet } \,.$$

Observe that

$$H_n( X[k]_\bullet) \simeq H_{n-k}(X_\bullet) \,.$$

It is therefore sufficient to check that

$$H_n \left( \array{ cone(f)_\bullet &\to& X[1]_\bullet \\ \downarrow^{\mathrlap{\simeq}} \\ Z_\bullet } \right) \;\; : \;\; H_n(Z_\bullet) \to H_n(cone(f)_\bullet) \to H_{n-1}(X_\bullet)$$

equals the connecting homomorphism $\delta_n$ induced by the short exact sequence.

By prop. the inverse of the vertical map is given by choosing lifts and forming the corresponding element given by the connecting homomorphism. By prop. the horizontal map is just the projection, and hence the assignment is of the form

$$[z_n] \mapsto [x_{n-1}, y_n] \mapsto [x_{n-1}] \,.$$

So in total the image of the zig-zag under homology sends

$$[z_n]_Z \mapsto -[\partial^Y \hat z_n]_X \,.$$

By the discussion there, this is indeed the action of the connecting homomorphism.

Distinguished triangles from mapping cones

In summary, the above says that for every chain map $f_\bullet : X_\bullet \to Y_\bullet$ we obtain maps

$$X_\bullet \stackrel{f}{\to} Y_\bullet \stackrel{ \left( \array{ 0 \\ id_{Y_\bullet} } \right) }{\to} cone(f)_\bullet \stackrel{ \left( \array{ id_{X[1]_\bullet} & 0 } \right) }{\to} X[1]_\bullet$$

which form a homotopy fiber sequence and such that this sequence continues by forming suspensions, hence for all $n \in \mathbb{Z}$ we have

$$X[n]_\bullet \stackrel{f}{\to} Y[n]_\bullet \stackrel{ \left( \array{ 0 \\ id_{Y[n]_\bullet} } \right) }{\to} cone(f)[n]_\bullet \stackrel{ \left( \array{ id_{X[n+11]_\bullet} & 0 } \right) }{\to} X[n+1]_\bullet$$

To amplify this quasi-cyclic behaviour one sometimes depicts the situation as follows:

$$\array{ X_\bullet &&\stackrel{f}{\to}&& Y_\bullet \\ & {}_{\mathllap{[1]}}\nwarrow && \swarrow \\ && cone(f)_\bullet }$$

and hence speaks of a “triangle”, or distinguished triangle or mapping cone triangle of $f$.

• distinguished triangle = period of homotopy fiber sequence .

Due to these “triangles” one calls the homotopy category of chain complexes localized at the quasi-isomorphisms, hence the derived category, a triangulated category.

Notice that equivalently we can express the triangles via the mapping cylinder. For every map of chain complexes $f:A\to B$, 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

$$A\to Cyl(u)\to Cone(u)\to A[1]$$

for some $u:A\to B$ where all the morphisms in the triangle are appropriatedly induced by $u$.

Related concepts

examples of universal constructions of topological spaces:

$\phantom{AAAA}$limits	$\phantom{AAAA}$colimits
$\,$ point space$\,$	$\,$ empty space $\,$
$\,$ product topological space $\,$	$\,$ disjoint union topological space $\,$
$\,$ topological subspace $\,$	$\,$ quotient topological space $\,$
$\,$ fiber space $\,$	$\,$ space attachment $\,$
$\,$ mapping cocylinder, mapping cocone $\,$	$\,$ mapping cylinder, mapping cone, mapping telescope $\,$
	$\,$ cell complex, CW-complex $\,$

References

In the context of chain complexes the construction is discussed for instance in

• Pierre Schapira, section 3.2 and section 7 of Categories and homological algebra (2011) (pdf)

• Charles Weibel, section 1.5 of An Introduction to Homological Algebra .

In the context of spectra discussion includes

• Robert Switzer, around 8. 17 of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

Graphics taken from

• Fernando Muro, Representability of Cohomology Theories, Joint Mathematical Conference CSASC 2010, 22–27 January 2010, Prague, Czech Republic (slides)