homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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Given a continuous map $f:X\to Y$ of topological spaces, one can define its mapping cylinder as a pushout
in Top, where $I = [0,1]$ (the unit interval) and $\sigma_0:X\to X\times I$ is given by $x\mapsto (x,0)$. By tradition, homotopy theorists sometimes use the inverted (upside-down) mapping cylinder where $\sigma_0$ is replaced by $\sigma_1:x\mapsto (x,1)$. Of course the two mapping cylinders are homeomorphic so it is matter of convention which one to use, of course, compatibly with other constructions depending on the orientation of $I$.
Set-theoretically, the mapping cylinder is usually represented as the quotient space $(X\times I \coprod Y)/{\sim}$ where $\sim$ is the smallest equivalence relation identifying $(x,0)\sim f(x)$ for all $x\in X$.
As any other pushout, the mapping cylinder has a universal property: for any space $Z$ and mapping $g_1:X\times I\to Z$, $g_2:Y\to Z$ such that $g_1(x,0)=g_2(f(x))$ for all $x\in X$, there is a unique $k:Cyl(f)\to Z$, such that the composition $X\times I\to Cyl(f)\stackrel{k}\to Z$ equals $g_1$ and the composition $Y\to Cyl(f)\stackrel{k}\to Z$ equals $g_2$.
For $f \colon X\to Y$ a continuous function, the canonical map $j \coloneqq f_*(\sigma_0) \colon Y\to Cyl(f)$ is a homotopy equivalence. In fact its homotopy inverse can be chosen a deformation retraction. In particular every continuous function factors as a map into its mapping cylinder followed by a deformation retraction.
We exhibit $j$ as a homotopy equivalence by constructing its homotopy inverse $\tilde{f}$ given by $\tilde{f}:[x,t]\mapsto f(x)$, where $[x,t]$ is a class of $(x,t)\in X\times I$ and $\tilde{f}([y])=[y]$ for $y\in Y$. Clearly this map is well-defined and $\tilde{f}\circ j = \id_Y$. On the other hand, $(j\circ\tilde{f})[x,t] = [f(x)]$. Homotopy $H:\mathrm{Cyl}(f)\times I\to Y$ is given by
It is easy to see that $H(-,0) = \id_{Cyl(f)}$, $H(-,1)=[-,0]=[f(-)]$ hence $j\circ\tilde{f}\sim id_{Cyl(f)}$.
A continuous map $i:A\to X$ is a Hurewicz cofibration iff there is a retraction $r:X\times I\to Cyl(f)$ for the canonical map $X\times I \to Cyl(f)$.
A continuous map $f:X\to Y$ is a homotopy equivalence iff $X = X\times\{0\}$ is a deformation retract of the cylinder $Cyl(f)$.
For any $f:X\to Y$, the composition
is a Hurewicz cofibration. Furthermore, the map $r:Cyl(f)\to Y$ determined by $r([x,t])= f(x)$ (for all $x\in X$ and $t\in I$) and $r([y])=y$ (for $y\in Y$) is well defined and a homotopy equivalence.
The composition $r\circ (\sigma_0)_* (f)\circ \sigma_1 = f$, hence this is a decomposition of a continuous map into a cofibration followed by a homotopy equivalence.
In homotopy type theory mapping cyclinders can be constructed as higher inductive types. See there.
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 $\,$ |
(…)
Discussion of mapping cylinders of topological spaces in general topology:
See also:
Discussion of mapping cylinders of chain complexes in homological algebra:
Textbook accounts:
Lecture notes: