# nLab cocylinder

Cocylinders and mapping cocylinders

### Context

#### Limits and colimits

limits and colimits

# Cocylinders and mapping cocylinders

## Ideas

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 $X^I$ or a path space object. In general, however, a cocylinder, $X^I$, may not involve any object $I$ 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 $X$, 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 space 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:

###### Definition

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

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

where $Y^I$ is the cocylinder.

The mapping cocylinder is sometimes denoted $M_f Y$ or $N f$.

###### Remark

If we interchange $ev_0$ and $ev_1$ 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.

### In type theory

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

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

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

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

## Examples

• In the case of topological spaces, the mapping cocylinder is the subspace $Cocyl(f)\subset Y^I\times X$ 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 $y\colon Y \vdash hfiber(f,y)\colon Type$. This is used crucially in the definition of equivalence in homotopy type theory.

## Applications

examples of universal constructions of topological spaces:

$\phantom{AAAA}$limits$\phantom{AAAA}$colimits
$\,$ 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 $\,$

## References

Peter May’s books use the terminology mapping path space.

Last revised on June 18, 2018 at 20:04:59. See the history of this page for a list of all contributions to it.