nLab
mapping cone

Idea

The mapping cone of a morphism $f:X\to Y$ is a particular model category representative of the homotopy cofiber of $f$, also called the homotopy cokernel of $f$ or weak quotient of $Y$ by the image of $X$ in $Y$ under $f$.

The dual notion is that of mapping cocone.

Definition

In an (∞,1)-category the homotopy cofiber of a morphism $f:X\to Y$ is the homotopy pushout

$$\begin{array}{ccc}X& \stackrel{}{\to }& *\\ {\downarrow }^f& & \downarrow \\ Y& \to & \mathrm{coker}(f)\end{array}.$$\array{ X &\stackrel{}{\to}& {*} \\ \downarrow^f && \downarrow \\ Y &\to& coker(f) } \,.

When the (∞,1)-category is presented by a category of cofibrant objects (for instance a model category with only cofibrant objects) then this may be computed by the ordinary colimit

$$\begin{array}{ccccc}& & X& \stackrel{f}{\to }& Y\\ & & {\downarrow }^{i_1}& & \downarrow \\ X& \stackrel{i_0}{\to }& \mathrm{Cyl}(X)\\ \downarrow & & & \searrow & \downarrow \\ *& \to & & \to & \mathrm{coker}(f)\end{array}$$\array{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{i_1} && \downarrow \\ X &\stackrel{i_0}{\to}& Cyl(X) \\ \downarrow && &\searrow & \downarrow \\ {*} &\to& &\to& coker(f) }

using a cylinder object $\mathrm{Cyl}(X)$ for $X$, that models the left homotopy filling the original homotopy pushout diagram. This colimit, in turn, may be computed in two stages by two consecutive pushouts as

$$\begin{array}{ccccc}& & X& \stackrel{f}{\to }& Y\\ & & {\downarrow }^{i_1}& & \downarrow \\ X& \stackrel{i_0}{\to }& \mathrm{Cyl}(X)\\ \downarrow & & \downarrow \\ *& \to & \mathrm{cone}(X)& \to & \mathrm{coker}(f)\end{array}.$$\array{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{i_1} && \downarrow \\ X &\stackrel{i_0}{\to}& Cyl(X) \\ \downarrow && \downarrow \\ {*} &\to& cone(X) &\to& coker(f) } \,.

The first pushout here

$$\begin{array}{ccc}X& \stackrel{i_0}{\to }& \mathrm{Cyl}(X)\\ \downarrow & & \downarrow \\ *& \to & \mathrm{cone}(X)\end{array}$$\array{ X &\stackrel{i_0}{\to}& Cyl(X) \\ \downarrow && \downarrow \\ {*} &\to& cone(X) }

is the cone over $X$: the result of taking the cylinder over $X$ and identifying one $X$-shaped end with the point.

The remaining pushout

$$\begin{array}{ccc}X& \stackrel{f}{\to }& Y\\ \downarrow & & \downarrow \\ \mathrm{cone}(X)& \to & \mathrm{coker}(f)& =:\mathrm{cone}(f)\end{array}$$\array{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow \\ cone(X) &\to& coker(f) & =: cone(f) }

is the mapping cone of $f$:

this is the result of taking that other remaining end of the cyclinder and gluing that to $Y$, using the identification given by $f$.

The geometric intuition behind this is best seen in the archetypical example of the model category Top. See the examples below.

Examples

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 $X$.

In Top

The construction is geometrically most obvious in the category Top of topological spaces.

Here for $I=[\mathrm{0,1}]$ the standard interval object we may take the cylinder object to be $\mathrm{Cyl}(X)=X\times I$, literally the cylinder over $X$.

Given a continuous map $f:X\to Y$, the topological space $\mathrm{cone}(f)$ is

$$X\times I{\cup }_{f}Y$$X\times I \cup_{f} Y

(here for all $x\in X$, $(x\mathrm{,1})$ is identified with $f(x)$) modulo the contraction of $X\times \{0\}$ to a point. The opposite convention is also possible: identify $(x\mathrm{,0})$ 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”.

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).

In additive categories with translation

Let $C$ be an additive category with translation $T=[1]:C\to C$. Let $X$ and $Y$ be two differential objects in $(C,T)$ and $f:X\to Y$ any morphism in $C$.

The mapping cone $\mathrm{Cone}(f)$ of $f$ is the differential object whose underlying object is the direct sum $TX\oplus Y$ and whose differential $d_{\mathrm{cone}f}:TX\oplus X\to TTX\oplus TX$ is given in matrix calculus notation by

$$d_{\mathrm{cone}f}:=\left(\begin{array}{cc}{d}_{TX}& 0\\ T(f)& d_Y\end{array}\right)=\left(\begin{array}{cc}-T(d_X)& 0\\ T(f)& d_Y\end{array}\right).$$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.

Distinguished triangles from mapping cones

A homotopy category of the category of chain complexes (with respect to chain homotopy equivalences) has a natural structure of a triangulated category where the distinguished triangles are the triangles isomorphic to mapping cone triangles

$$A\stackrel{f}{\to }B\stackrel{\left(\begin{array}{c}0\\ {\mathrm{Id}}_B\end{array}\right)}{\to }\mathrm{Cone}(f)\stackrel{\left(\begin{array}{cc}{\mathrm{Id}}_{A[1]}& 0\end{array}\right)}{\to }A[1].$$A \stackrel{f}{\to} B \stackrel{ \left( \array{ 0 \\ Id_B } \right) }{\to} Cone(f) \stackrel{ \left( \array{ Id_{A[1]} & 0 } \right) }{\to} A[1] \,.

For every map of chain complexes $f:A\to B$, the cylinder $\mathrm{Cyl}(f)$ 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

$$A\to \mathrm{Cyl}(u)\to \mathrm{Cone}(u)\to A[1]$$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 $u$.