Contents

Idea

The mapping cone of a morphism in some homotopical category (precisely: a category of cofibrant objects) is, if it exists, a particular representative of the homotopy cofiber of .

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

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.

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

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

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 to the terminal object is the suspension object of an object . The dual notion of the loop space object of .

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 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 to be

which is literally the cylinder over .

Given a continuous function , the topological space is

This is the disjoint union of with followed by an identification under which for each a point is identified with the point and followed by the contraction of to a point.

Of course the opposite convention is also possible: identify with for all and then contract 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 be the category of chain complexes in Mod for some ring .

(For instance if the integers, then this is , chain complexes of abelian groups. More generally 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 of a chain map , according to the general definition . The end result is prop. below, reproducing the classical formula for the mapping cone.

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

One part of definition now reads:

The last part of definition now reads:

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

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

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 be some concrete additive category and the category of chain complexes in . For

a morphism, the mapping cone is the complex

There is a canonical cochain homotopy

where is the canonical inclusion, componentwise given by

and where the cochain homotopy has components

which we denote on by

The fact that this is a cochain homotopy means that

which we check on any by computing

where we used the above definition of and the fact that is a chain homomorphism and hence intertwines the differentials.

This cochain homotopy is universal in that for any other cochain homotopy

hence

we have a morphism

given on by and on by

which is indeed a cochain homomorphism because for all we have

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

hence that

and

In additive categories with translation

Let be an additive category with translation . Let and be two differential objects in and any morphism in .

The mapping cone of is the differential object whose underlying object is the direct sum and whose differential is given in matrix calculus notation by

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 be a chain map and write for its mapping cone as explicitly given in prop. .

In order to compare this to the discussion of connecting homomorphisms, we now turn attention to the case that happens to be a monomorphism. Notice that this we can always assume, up to quasi-isomorphism, for instance by prolonging by the map into its mapping cylinder

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

of chain complexes. The following discussion revolves around the fact that now as well as are both models for the homotopy cofiber of .

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

Distinguished triangles from mapping cones

In summary, the above says that for every chain map we obtain maps

which form a homotopy fiber sequence and such that this sequence continues by forming suspensions, hence for all we have

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

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

• 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 , the cylinder is quasi-isomorphic to , and moreover in the homotopy category of chain complexes, every distinguished triangle is quasi-isomorphic to a distinguished triangle of the form

for some where all the morphisms in the triangle are appropriatedly induced by .

Related concepts

References

In the context of chain complexes:

• Masaki Kashiwara, Pierre Schapira, sections 4.2 & 7 of: An Introduction to Categories and Homological Algebra (2024) [pdf]

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

In the context of spectra:

• 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.

The above graphics is taken from

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