Limits and colimits

Homotopy theory

Cocylinders and mapping cocylinders


In algebraic topology and homotopy theory, a cocylinder is a dual construction to a cylinder. In contexts where spatial intuition is involved, it is perhaps more often called a path space X IX^I or a path space object. In general, however, a cocylinder, X IX^I, may not involve any object II nor use a mapping space in its construction, see cylinder functor for the discussion of the dual point.

Definition (cocylinders and cocylinder functors)

These are the duals of cylinders and cylinder functors so can safely be left as an exercise.

Ideas continued

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 (mapping cocylinders)

In category theory

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

In any of these cases:


Given a morphism f:XYf\colon X\to Y, its mapping cocylinder (or mapping path space or mapping path fibration) is the pullback

Cocyl(f) X f Y I ev 0 Y ev 1 Y \array{ Cocyl(f)&\to& X\\ \downarrow&&\downarrow f \\ Y^I&\stackrel{ev_0}{\to}&Y \\ \downarrow^{\mathrlap{ev_1}} \\ Y }

where Y IY^I is the cocylinder.

The mapping cocylinder is sometimes denoted M fYM_f Y or NfN f.


If we interchange ev 0ev_0 and ev 1ev_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.

In type theory

In homotopy type theory the mapping cocylinder Cocyl(f)YCocyl(f) \to Y is expressed as

y:Y xX(f(x)=y) y : Y \vdash \sum_{x \in X} (f(x) = y)

being the dependent sum over xx of the substitution of f(x)f(x) for y 1y_1 in the dependent identity type (y 1=y)(y_1 = y). Equivalently this is the yy-dependent homotopy fiber of ff at yy

y:Yhfiber(f,y). y : Y \vdash hfiber(f,y) \,.




(This uses the terminology mapping path space.)

Revised on May 16, 2012 15:27:27 by Tim Porter (