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\begin{document}

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\section*{cylinder object}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy Theory}}\label{homotopy_theory}

[[ !include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of a \textbf{cylinder object} in a [[category]] is an abstraction of the construction in [[Top]] which associates to any [[topological space]] $X$ the \emph{cylinder} $X \times [0,1]$ over $X$, where $[0,1]$ is the standard [[topological interval]]. It is notably used to define the concept of [[left homotopy]], say in a [[model category]].

The standard topological cylinder $X \times [0,1]$ naturally comes equipped with a continuous map

\begin{displaymath}
X \coprod X \to X \times [0,1]
\end{displaymath}
that identifies $X$ as the two ends $X \times \{0\}$ and $X \times \{1\}$ of the cylinder, and with a map

\begin{displaymath}
X \times [0,1] \to X
\end{displaymath}
that collapses the cylinder back onto $X$.

The composite of these two maps is the [[codiagonal]] $(Id,Id) :  X \coprod X \to X$. Moreover, the cylinder $X \times [0,1]$ is [[homotopy equivalence|homotopy equivalent]] to $X$.

These properties are the characterizing properties of the cylinder that can be abstracted and realized in other categories.

The notion dual to \emph{cylinder object} is [[path space object]], which is thus sometimes alternatively called a cocylinder. Cylinder objects and path space objects are used to define [[left homotopies]] and [[right homotopies]], respectively.

There are several views on the role of cylinders / cocylinders in homotopy theory. If there is a natural notion of weak equivalence or [[quasi-isomorphism]] then the cylinder is used to encode a notion of homotopy equivalence compatible with the weak equivalences. In some other situations, a `cylinder' , often functorially given and well structured in some way, may be the \emph{primitive} notion that allows a notion of `homotopy equivalence' to be put forward. Below we give a definition optimised for the former situation. Some indication of the second context is given in the entry [[cylinder functor]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In a [[category with weak equivalences]] $C$ that has [[coproducts]] a \textbf{cylinder object} $Cyl(X)$ for an [[object]] $X$ is a factorization

\begin{displaymath}
X \coprod X \to Cyl(X) \stackrel{\simeq}{\to} X
\end{displaymath}
of the codiagonal $X \coprod X \to X$ out of the [[coproduct]] of $X$ with itself, such that $Cyl(X) \to X$ is a weak equivalence and such that the morphism $X \coprod X \to Cyl(X)$ is ``nice'' in some way.

In some situations the assignment of cylinder objects may exist functorially, in which case one speaks of a [[cylinder functor]].

If $C$ has the structure of a [[model category]] then ``nice'' means that $X \coprod X \hookrightarrow Cyl(X)$ is a [[cofibration]]. The factorization axiom of a model category ensures that for each object there is a cylinder object with this property; in fact, one with the additional property that $Cyl(X) \to X$ is an acyclic fibration. Cylinder objects such that $X \coprod X \hookrightarrow Cyl(X)$ is a cofibration are sometimes called \emph{good}, and those for which moreover $Cyl(X) \to X$ is an acyclic fibration are then called \emph{very good}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item In [[sSet]] equipped with the standard [[model structure on simplicial sets]] the standard cylinder object for any $S \in sSet$ is $S \times \Delta[1]$.


\item In [[Top]], the standard cylinder $X\times [0,1]$ is a cylinder object for both the [[classical model structure on topological spaces]] $Top_{Quillen}$ (the one with [[Serre fibrations]]) as well as for the [[Strøm model structure]] $Top_{Strom}$ (the one with [[Hurewicz fibrations]]).

This standard cylinder is generally a ``good cylinder'' in the above sense only for $Top_{Stron}$ (in which case it is in fact a ``very good cylinder'').

In $Top_{Quillen}$ a sufficient condition for the standard cylinder $X\times I$ to be good is that $X$ is a [[CW-complex]].



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[cylinder functor]]


\item [[left homotopy]]


\item [[mapping cylinder]]


\item [[path space object]]


\item [[reduced cylinder]]


\item [[cylinder spectrum]]


\item [[h-cobordism]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The precise argument that for $X$ a [[cell complex]] then also the standard cyclinder $X\times I$ is a cell complex is spelled out for instance as prop. 2.9 in

\begin{itemize}%
\item Ottina, \emph{An A-based cofibrantly generated model category} (\href{http://arxiv.org/abs/1405.2086}{arXi:1405.2086})

\end{itemize}
[[!redirects cylinder objects]] [[!redirects cylinder]] [[!redirects cylinders]]

[[!redirects good cylinder object]] [[!redirects good cylinder objects]]



\end{document}