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\begin{document}

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\section*{cocylinder}

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

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

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

[[!include homotopy - contents]]

\hypertarget{cocylinders_and_mapping_cocylinders}{}\section*{{Cocylinders and mapping cocylinders}}\label{cocylinders_and_mapping_cocylinders}

\noindent\hyperlink{ideas}{Ideas}\dotfill \pageref*{ideas} \linebreak
\noindent\hyperlink{definition_cocylinders_and_cocylinder_functors}{Definition (cocylinders and cocylinder functors)}\dotfill \pageref*{definition_cocylinders_and_cocylinder_functors} \linebreak
\noindent\hyperlink{ideas_continued}{Ideas continued}\dotfill \pageref*{ideas_continued} \linebreak
\noindent\hyperlink{definition_mapping_cocylinders}{Definition (mapping cocylinders)}\dotfill \pageref*{definition_mapping_cocylinders} \linebreak
\noindent\hyperlink{in_category_theory}{In category theory}\dotfill \pageref*{in_category_theory} \linebreak
\noindent\hyperlink{in_type_theory}{In type theory}\dotfill \pageref*{in_type_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{ideas}{}\subsection*{{Ideas}}\label{ideas}

In [[algebraic topology]] and [[homotopy theory]], a \textbf{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.

\hypertarget{definition_cocylinders_and_cocylinder_functors}{}\subsection*{{Definition (cocylinders and cocylinder functors)}}\label{definition_cocylinders_and_cocylinder_functors}

These are the duals of [[cylinders]] and [[cylinder functors]] so can safely be left as an exercise.

\hypertarget{ideas_continued}{}\subsection*{{Ideas continued}}\label{ideas_continued}

Similarly, the \textbf{mapping cocylinder}, which is dual to the [[mapping cylinder]], is equally called the \textbf{mapping path space} or \textbf{mapping path fibration}. It provides a canonical way to factor any map as a [[homotopy equivalence]] followed by a [[fibration]].

\hypertarget{definition_mapping_cocylinders}{}\subsection*{{Definition (mapping cocylinders)}}\label{definition_mapping_cocylinders}

\hypertarget{in_category_theory}{}\subsubsection*{{In category theory}}\label{in_category_theory}

For a [[topological space]] $X$, its \textbf{cocylinder} is simply the path space $X^{[0,1]}$. More generally, in a [[cartesian closed category]] with an [[interval object]] $I$, the \textbf{cocylinder} of an object $X$ is the [[exponential object]] $X^I$. Even more generally, in a [[model category]] the \textbf{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:

\begin{defn}
\label{}\hypertarget{}{}
Given a [[morphism]] $f\colon X\to Y$, its \textbf{mapping cocylinder} (or \textbf{mapping path space} or \textbf{mapping path fibration}) is the [[pullback]]

\begin{displaymath}
\itexarray{
   Cocyl(f)&\to& X\\
    \downarrow&&\downarrow f \\
    Y^I&\stackrel{ev_0}{\to}&Y
    \\
    \downarrow^{\mathrlap{ev_1}}
    \\
    Y
 }
\end{displaymath}
where $Y^I$ is the cocylinder.

\end{defn}
The mapping cocylinder is sometimes denoted $M_f Y$ or $N f$.

\begin{remark}
\label{}\hypertarget{}{}
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.

\end{remark}
\hypertarget{in_type_theory}{}\subsubsection*{{In type theory}}\label{in_type_theory}

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

\begin{displaymath}
y : Y \vdash \sum_{x  \in X} (f(x) = y)
\end{displaymath}
being the [[dependent sum]] over $x$ of the [[substitution]] of $f(x)$ for $y_1$ in the [[dependent type|dependent]] [[identity type]] $(y_1 = y)$. Equivalently this is the $y$-[[dependent type|dependent]] [[homotopy fiber]] of $f$ at $y$

\begin{displaymath}
y : Y \vdash hfiber(f,y)
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item 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 $s(0)=f(x)$.


\item 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]].



\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item The mapping cocylinder is the central ingredient in the [[factorization lemma]].


\item One usage is discussed at \emph{[[Hurewicz connection]]}.


\item The mapping path fibration is used in the construction of the [[Strøm model structure]] on topological spaces.


\item The [[homotopy fiber]] can be constructed as the strict [[fiber]] of the mapping cocylinder.



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

[[!include universal constructions of topological spaces -- table]]

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

\begin{itemize}%
\item [[George Whitehead]], \emph{Elements of homotopy theory}

\end{itemize}
Peter May's books use the terminology \emph{mapping path space}.

[[!redirects cocylinder]] [[!redirects cocylinders]] [[!redirects mapping cocylinder]] [[!redirects mapping cocylinders]] [[!redirects mapping path space]] [[!redirects mapping path spaces]] [[!redirects mapping path fibration]] [[!redirects mapping path fibrations]]



\end{document}