nLab
cocylinder

Idea

In algebraic topology and homotopy theory, a cocylinder is a dual construction to a cylinder. However, it is more commonly called a path space $X^I$ or a path object.

Similarly, the mapping cocylinder, which is dual to the mapping cylinder, is equally called the mapping path space or mapping path fibration. It provides a canonical way to factor any map as a homotopy equivalence followed by a fibration.

Definition

For a topological space $X$, its cocylinder is simply the path space $X^{[0,1]}$. More generally, in a cartesian closed category with an interval object $I$, the cocylinder of an object $X$ is the exponential object $X^I$. Even more generally, in a model category the cocylinder of any object is the path object — the factorization of the diagonal morphism $X\to X\times X$ as an acyclic cofibration followed by a fibration.

In any of these cases, given a morphism $f:X\to Y$, its mapping cocylinder (or mapping path space or mapping path fibration) is the pullback

$$\begin{array}{ccc}\mathrm{Cocyl}(f)& \to & X\\ \downarrow & & \downarrow f\\ Y^I& \stackrel{{\mathrm{ev}}_0}{\to }& Y\end{array}$$\array{ Cocyl(f)&\to& X\\ \downarrow&&\downarrow f \\ Y^I&\stackrel{ev_0}{\to}&Y }

where $Y^I$ is the cocylinder. The mapping cocylinder is sometimes denoted $M_{f}Y$ or $Nf$.

Examples

• In the case of topological spaces, the mapping cocylinder is the subspace $\mathrm{Cocyl}(f)\subset Y^I\times X$ whose elements are pairs $(s,x)$ such that $s(0)=f(x)$.

• In homotopy type theory, cocylinders represent identity types, and the mapping cocylinder represents the dependent type $y:Y⊢\mathrm{hfiber}(f,y):\mathrm{Type}$. This is used crucially in the definition of equivalence in homotopy type theory.

Applications

• For a usage see Hurewicz connection.

• In Brown’s theory of higher stack?s via categories of fibred objects, mapping cocylinders take a role of total spaces of a relative version of universal principal bundles? associated to a map $f$; the projection of such a bundle is the composition $\mathrm{Cocyl}(f)\to Y^I\stackrel{{\mathrm{ev}}_1}{\to }Y$. Note that the other leg ${\mathrm{ev}}_1$ is used here.

• The mapping path fibration is used in the construction of the Strøm model structure on topological spaces.

• The homotopy fiber can be constructed as the strict fiber of the mapping cocylinder.

Remarks

• If we interchange ${\mathrm{ev}}_0$ and ${\mathrm{ev}}_1$ then we have an upside-down version of a cylinder, sometimes called inverse (or inverted) mapping cocylinder; but usually it is clear just from the context which version is used. They are homotopy equivalent, so usually it does not matter.

References

• George Whitehead in his old book “Elements of homotopy theory” uses the terminology mapping path space.