Limits and colimits

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


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) \,.



examples of universal constructions of topological spaces:

\, 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 \,


(This uses the terminology mapping path space.)

Revised on May 2, 2017 13:14:41 by Urs Schreiber (