nLab
cocylinder

In algebraic topology a cocylinder is a dual construction to a cylinder or a mapping cylinder.

The cocylinder $\mathrm{Cocyl}(X)$ of a space $X$ is simply the path space $X^I$, see entry path object. The terminology is however rather used in its extension to the mappings where the path terminology is less often used: a (mapping) cocylinder of a continuous map $f:X\to Y$ 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 path object. In other words, that is the subspace $\mathrm{Cocyl}(f)\subset Y^I\times X$ whose elements are pairs $(s,x)$ such that $s(0)=f(x)$. For a usage see Hurewicz connection. George Whitehead in his old book “Elements of homotopy theory” instead uses the terminology mapping path space with notation $M_{f}Y$. 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.

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.