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\begin{document}

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\section*{homotopy}

\begin{quote}%
This page is about homotopy as a transformation. For homotopy sets in [[homotopy categories]], see [[homotopy (as an operation)]].


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\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{in_topological_spaces}{In topological spaces}\dotfill \pageref*{in_topological_spaces} \linebreak
\noindent\hyperlink{in_enriched_categories}{In enriched categories}\dotfill \pageref*{in_enriched_categories} \linebreak
\noindent\hyperlink{in_model_categories}{In model categories}\dotfill \pageref*{in_model_categories} \linebreak
\noindent\hyperlink{in_cofibration_categories}{In (co-)fibration categories}\dotfill \pageref*{in_cofibration_categories} \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}

In many [[category|categories]] $C$ in which one does [[homotopy theory]], there is a notion of \emph{homotopy} between [[morphisms]], which is closely related to the [[2-morphisms]] in [[higher category theory]]: a homotopy between two morphisms is a way in which they are equivalent.

If we regard such a category as a presentation of an $(\infty,1)$-[[(infinity,1)-category|category]], then homotopies $f\sim g$ present the 2-cells $f\Rightarrow g$ in the resulting $(\infty,1)$-category.

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

\hypertarget{in_topological_spaces}{}\subsubsection*{{In topological spaces}}\label{in_topological_spaces}

\begin{defn}
\label{LeftHomotopy}\hypertarget{LeftHomotopy}{}
For $f,g\colon X \longrightarrow Y$ two [[continuous functions]] between [[topological spaces]] $X,Y$, then a \textbf{left homotopy}

\begin{displaymath}
\eta \colon f \,\Rightarrow_L\, g
\end{displaymath}
is a [[continuous function]]

\begin{displaymath}
\eta \;\colon\; X \times I \longrightarrow Y
\end{displaymath}
out of the standard [[cylinder object]] over $X$: the [[product space]] of $X$ with the [[Euclidean space|Euclidean]] [[closed interval]] $I \coloneqq [0,1]$, such that this fits into a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
     X
     \\
     {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}}
     \\
     X \times I &\stackrel{\eta}{\longrightarrow}& Y
     \\
     {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}}
     \\
     X
  }
  \,.
\end{displaymath}
(graphics grabbed from J. Tauber \href{http://jtauber.com/blog/2005/07/01/path_homotopy/}{here})

\end{defn}
\begin{example}
\label{PathsAsLeftHomotopyBetweenPoints}\hypertarget{PathsAsLeftHomotopyBetweenPoints}{}
Let $X$ be a [[topological space]] and let $x,y \in X$ be two of its points, regarded as functions $x,y \colon \ast \longrightarrow X$ from the point to $X$. Then a left homotopy, def. \ref{LeftHomotopy}, between these two functions is a commuting diagram of the form

\begin{displaymath}
\itexarray{
     \ast
     \\
     {}^{\mathllap{\delta_0}}\downarrow & \searrow^{\mathrlap{x}}
     \\
     I &\stackrel{\eta}{\longrightarrow}& Y
     \\
     {}^{\mathllap{\delta_1}}\uparrow & \nearrow_{\mathrlap{y}}
     \\
     \ast
  }
  \,.
\end{displaymath}
This is simply a continuous path in $X$ whose endpoints are $x$ and $y$.

\end{example}
\hypertarget{in_enriched_categories}{}\subsubsection*{{In enriched categories}}\label{in_enriched_categories}

If $C$ is [[enriched category|enriched]] over [[Top]], then a \textbf{homotopy} in $C$ between maps $f,g:X\,\rightrightarrows \,Y$ is a map $H:[0,1] \to C(X,Y)$ in $Top$ such that $H(0)=f$ and $H(1)=g$. In $Top$ itself this is the classical notion.

If $C$ has [[copower|copowers]], then an equivalent definition is a map $[0,1]\odot X\to Y$, while if it has [[power|powers]], an equivalent definition is a map $X\to \pitchfork([0,1],Y)$.

There is a similar definition in a [[simplicially enriched category]], replacing $[0,1]$ with the 1-simplex $\Delta^1$, with the caveat that in this case not all \emph{simplicial homotopies} need be composable even if they match correctly. (This depends on whether or not all (2,1)-[[horn]]s in the simplicial set, $C(X,Y)$, have fillers.) Likewise in a [[dg-category]] we can use the ``chain complex interval'' to get a notion of \emph{chain homotopy}.

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

If $\mathcal{C}$ is a [[model category]], it has an intrinsic notion of homotopy determined by its factorizations. For more on the following see at \emph{[[homotopy in a model category]]}.

\begin{defn}
\label{PathAndCylinderObjectsInAModelCategory}\hypertarget{PathAndCylinderObjectsInAModelCategory}{}
Let $\mathcal{C}$ be a [[model category]] and $X \in \mathcal{C}$ an [[object]].

\begin{itemize}%
\item A \textbf{[[path object]]} $Path(X)$ for $X$ is a factorization of the [[diagonal]] $\nabla_X \colon X \to X \times X$ as

\end{itemize}
\begin{displaymath}
\nabla_X 
  \;\colon\;
   X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X
  \,.
\end{displaymath}
where $X\to Path(X)$ is a weak equivalence. This is called a \textbf{good path object} if in addition $Path(X) \to X \times X$ is a fibration.

\begin{itemize}%
\item A \textbf{[[cylinder object]]} $Cyl(X)$ for $X$ is a factorization of the [[codiagonal]] (or ``fold map'') $\Delta_X: X \sqcup X \to X$ as

\end{itemize}
\begin{displaymath}
\Delta_X
  \;\colon\;
  X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X
  \,.
\end{displaymath}
where $Cyl(X) \to X$ is a weak equivalence. This is called a \textbf{good cylinder object} if in addition $X \sqcup X \to Cyl(X)$ is a cofibration.

\end{defn}
\begin{remark}
\label{RemarkOnChoicesOfNonGoodPathAndCylinderObjects}\hypertarget{RemarkOnChoicesOfNonGoodPathAndCylinderObjects}{}
By the factorization axioms every object in a model category has both a good path object and as well as a good cylinder object according to def. \ref{PathAndCylinderObjectsInAModelCategory}. But in some situations one is genuinely interested in using non-good such objects.

For instance in the [[classical model structure on topological spaces]], the obvious object $X\times [0,1]$ is a cylinder object, but not a good cylinder unless $X$ itself is cofibrant (a [[cell complex]] in this case).

More generally, the path object $Path(X)$ of def. \ref{PathAndCylinderObjectsInAModelCategory} is analogous to the [[powering]] $\pitchfork(I,X)$ with an [[interval object]] and the cyclinder object $Cyl(X)$ is analogous to the [[tensoring]] with a cylinder object $I\odot X$. In fact, if $\mathcal{C}$ is a $V$-[[enriched model category]] and $X$ is fibrant/cofibrant, then these powers and copowers are in fact examples of (good) path and cylinder objects if the [[interval object]] is sufficiently good.

\end{remark}
\begin{defn}
\label{LeftAndRightHomotopyInAModelCategory}\hypertarget{LeftAndRightHomotopyInAModelCategory}{}
Let $f,g \colon X \longrightarrow Y$ be two [[parallel morphisms]] in a [[model category]].

\begin{itemize}%
\item A \textbf{left homotopy} $\eta \colon f \Rightarrow_L g$ is a morphism $\eta \colon Cyl(X) \longrightarrow Y$ from a [[cylinder object]] of $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, such that it makes this [[commuting diagram|diagram commute]]:

\end{itemize}
\begin{displaymath}
\itexarray{
    X &\longrightarrow& Cyl(X) &\longleftarrow& X
    \\
    & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}}
    \\
    && 
    Y
  }
  \,.
\end{displaymath}
\begin{itemize}%
\item A \textbf{right homotopy} $\eta \colon f \Rightarrow_R g$ is a morphism $\eta \colon X \to Path(Y)$ to some [[path object]] of $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, such that this [[commuting diagram|diagram commutes]]:

\end{itemize}
\begin{displaymath}
\itexarray{
    && X
    \\
    & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} 
    \\
    Y &\longleftarrow& Path(Y) &\longrightarrow& Y
  }
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
By remark \ref{RemarkOnChoicesOfNonGoodPathAndCylinderObjects} it follows that in a $Top$-[[enriched model category]], any enriched homotopy between maps $X\to Y$ is a left homotopy if $X$ is cofibrant and a right homotopy if $Y$ is fibrant. Similar remarks hold for other enrichments.

\end{remark}
For more see at \emph{[[homotopy in a model category]]}.

\hypertarget{in_cofibration_categories}{}\subsubsection*{{In (co-)fibration categories}}\label{in_cofibration_categories}

Clearly the concepf of left homotopy in def. \ref{PathAndCylinderObjectsInAModelCategory} only needs part of the model category axioms and thus makes sense more generally in suitable [[cofibration categories]]. Dually, the concepf of path ojects in def. \ref{PathAndCylinderObjectsInAModelCategory} makes sense more generally in suitable [[fibration categories]] such as [[categories of fibrant objects]] in the sense of Brown.

Likewise if there is a [[cylinder functor]], one gets functorially defined [[cylinder objects]], etc.

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

[[!include homotopy-homology-cohomology]]

\begin{itemize}%
\item [[left homotopy]], [[right homotopy]]


\item [[homotopy relative boundary]]


\item [[isotopy]], [[smooth isotopy]]


\item [[higher homotopy]]


\item [[homotopy class]]


\item [[transfor]]

\begin{itemize}%
\item [[natural transformation]]


\item [[pseudonatural transformation]]


\item [[lax natural transformation]]



\end{itemize}

\item [[function extensionality]]



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

See the references at \emph{[[homotopy theory]]} and at \emph{[[model category]]}.

[[!redirects left homotopy]] [[!redirects right homotopy]]

[[!redirects left homotopies]] [[!redirects right homotopies]]

[[!redirects homotopy (as a transformation)]]

[[!redirects homotopies]]

[[!redirects homotopic]]



\end{document}