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\begin{document}

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\section*{path space object}

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

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

[[!include homotopy - contents]]

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - 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{in_model_categories}{In model categories}\dotfill \pageref*{in_model_categories} \linebreak
\noindent\hyperlink{in_simplicial_model_categories}{In simplicial model categories}\dotfill \pageref*{in_simplicial_model_categories} \linebreak
\noindent\hyperlink{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{right_homotopies}{Right homotopies}\dotfill \pageref*{right_homotopies} \linebreak
\noindent\hyperlink{loop_space_objects}{Loop space objects}\dotfill \pageref*{loop_space_objects} \linebreak


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

A \emph{path space object} in [[homotopy theory]] is an object that behaves for many purposes as the [[topology|topological]] [[path space]] in [[topological homotopy theory]].

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

In $C$ a [[category with weak equivalences]] and with [[products]] a \textbf{path space object of an object $X$} is a factorization of the diagonal morphism $X \stackrel{(Id, Id)}{\to} X \times X$ into the [[product]] as

\begin{displaymath}
X \stackrel{s}{\to} X^I \stackrel{(d_0, d_1)}{\to} X \times X
\end{displaymath}
such that $s$ is a weak equivalence. (This also makes sense even if the product $X \times X$ doesn't exist.) We interpret a ([[generalised element|generalised]]) [[global element|element]] of $X^I$ as a [[path]] in $X$.

\begin{uremark}
Here $C^I$ is a primitive symbol. $I$ is \emph{not} assumed to be an object and $C^I$ is not assumed to be an [[closed category|internal hom]]. This is standard but somewhat abusive notation. It is supposed to remind us of the ``nice'' situation where the path object \emph{is} co-represented by an [[interval object]].

\end{uremark}
If the category in question also has a notion of [[fibration]]s, such as in a [[category of fibrant objects]] or in a [[model category]], the morphism $C^I \stackrel{(d_0, d_1)}{\to} C \times C$ in the definition of a path object is required to be a fibration.

Path space objects are in particular guaranteed to exist in any [[model category]].

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

\hypertarget{in_model_categories}{}\subsubsection*{{In model categories}}\label{in_model_categories}

If $C$ is a [[model category]] then the factorization axiom ensures that for every object $X \in C$ there is a factorization of the diagonal

\begin{displaymath}
X \stackrel{\simeq}{\to} X^I \stackrel{(d_0, d_1)}{\to} X \times X
\end{displaymath}
with the additional property that $X^I \to X \times X$ is a fibration.

If $X$ itself is fibrant, then the projections $X \times X \to X$ are fibrations and moreover by 2-out-of-3 applied to the diagram

\begin{displaymath}
\itexarray{
    && X^I
    \\
    & {}^{\mathllap{s}}\nearrow && \searrow^{\mathrlap{d_i}}
    \\
    X &&\stackrel{Id}{\to}&& X
  }
\end{displaymath}
are themselves weak equivalences $X^I \stackrel{\simeq}{\to} X$. This is a key property that implies the [[category of fibrant objects|factorization lemma]].

If moreover the [[small object argument]] applies in the model category $C$, then such factorizations, and hence path objects, may be chosen functorially: such that for each morphism $X \to Y$ the factorizations fit into a [[commuting diagram]]

\begin{displaymath}
\itexarray{
     X &\stackrel{\simeq}{\to} &X^I &\to & X \times X  
    \\
    \downarrow && \downarrow && \downarrow
    \\
    Y & \stackrel{\simeq}{\to} & Y^I &\to & Y \times Y  
  }
\end{displaymath}
\hypertarget{in_simplicial_model_categories}{}\subsubsection*{{In simplicial model categories}}\label{in_simplicial_model_categories}

If $C$ is a [[simplicial model category]], then the [[power]]ing over [[sSet]] can be used to explicitly construct functorial path objects for fibrant objects $X$: define $X \to X^I \to X \times X$ to be the [[power|powering]] of $X$ by the morphisms

\begin{displaymath}
\Delta[0] \coprod \Delta[0]
  \stackrel{d_0, d_1}{\hookrightarrow} \Delta[1]
  \stackrel{\simeq}{\to} 
  \Delta[0]
\end{displaymath}
in $sSet_{Quillen}$. Notice that the first morphism is a cofibration and the second a weak equivalence in the standard [[model structure on simplicial sets]] and that all objects are cofibrant.

Since by the axioms of an [[enriched model category]] the [[power|powering functor]]

\begin{displaymath}
(-)^{(-)} : sSet^{op} \times C \to C
\end{displaymath}
sends cofibrations and acyclic cofibrations in the first argument to fibrations and acyclic fibrations inif the second argument is fibrant, and since this implies by the [[category of fibrant objects|factorization lemma]] that it then also preserves weak equivalences between cofibrant objects, it follows that $X^{\Delta[1]}$ is indeed a path object with the extra property that also the two morphisms $X^{\Delta[1]} \to X$ are acyclic fibrations.

\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

\hypertarget{right_homotopies}{}\subsubsection*{{Right homotopies}}\label{right_homotopies}

Path objects are used to define a notion of [[right homotopy]] between morphisms in a category. Thus they capture aspects of [[higher category theory]] in a $1$-categorical context.

\hypertarget{loop_space_objects}{}\subsubsection*{{Loop space objects}}\label{loop_space_objects}

From a path space object may be derived [[loop space object]]s.

[[!redirects path object]] [[!redirects path objects]] [[!redirects path space object]] [[!redirects path space objects]] [[!redirects cocylinder object]] [[!redirects cocylinder objects]]

[[!redirects path fibration]] [[!redirects path fibrations]]



\end{document}