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\begin{document}

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\section*{homotopy category}

\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{for_simplicially_enriched_categories}{For simplicially enriched categories}\dotfill \pageref*{for_simplicially_enriched_categories} \linebreak
\noindent\hyperlink{for_categories_with_weak_equivalences}{For categories with weak equivalences}\dotfill \pageref*{for_categories_with_weak_equivalences} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \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}

Given a [[category with weak equivalences]] $(\mathcal{C},W)$, then its \emph{homotopy category} $Ho(\mathcal{C})$ is, if it exists, the result of universally forcing the [[weak equivalences]] to become actual [[isomorphisms]], also called the \emph{[[localization]]} at the weak equivalences

\begin{displaymath}
\mathcal{C} \longrightarrow Ho(\mathcal{C})= \mathcal{C}[W^{-1}]
  \,.
\end{displaymath}
The classical example is the category of [[topological spaces]] with weak equivalences those [[continuous functions]] which are \emph{[[homotopy equivalences]]} or \emph{[[weak homotopy equivalences]]}. The corresponding homotopy category is often referred to as ``the homotopy category'', by default, or the ``[[classical homotopy category]]'' for emphasis. This turns out to be equivalent to the category of topological spaces or (for weak homotopy equivalences) of just those [[homeomorphism|homeomorphic]] to [[CW-complexes]] with [[left homotopy]]-[[homotopy class|classes]] of continuous functions between them, whence the name ``homotopy category''.

The existence of a homotopy category, as well as tractable presentations of it typically require extra [[properties]] of the class of weak equivalences (such as that they admit a [[calculus of fractions]]) or even extra [[structure]] (such as [[fibration category]]/[[cofibration category]] structure, or full [[model category]] structure, or further enhancements of that to [[simplicial model category]] structures, etc). See at \emph{[[homotopy category of a model category]]} for more on this.

More generally, to every [[(∞,1)-category]] is associated a homotopy category, whose morphisms are literally the [[homotopy classes]] of the original morphisms. See at \emph{[[homotopy category of an (∞,1)-category]]} for more on this.

These two concepts of ``homotopy category'' are compatible: to a [[category with weak equivalences]] is associated, if it exists, an [[(∞,1)-category]] obtained by universally forcing the [[weak equivalences]] to become actual [[homotopy equivalences]], also called the \emph{[[simplicial localization]]} $L_W \mathcal{C}$ at the weak equivalences. The homotopy categories of $(\mathcal{C},W)$ and of $L_W \mathcal{C}$ coincide, which justifies the terminology ``homotopy category'' generally.

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

\hypertarget{for_simplicially_enriched_categories}{}\subsubsection*{{For simplicially enriched categories}}\label{for_simplicially_enriched_categories}

Given a simplicially enriched category $C$, we can form for each pair of objects, $x,y$, of objects of $C$, the set, $\pi_0C(x,y)$, of connected components of the `function space' $C(x,y)$. As $\pi_0$ preserves finite limits, this gives a category, denoted $\pi_0(C)$. As 1-simplices in $C(x,y)$ can be often interpreted as being homotopies, this category $\pi_0(C)$ is often called the \emph{homotopy category of $C$}, and then the notation $Ho(C)$ may be used.

This notions is closely related to the next, by using, say the [[hammock localisation]] of Dwyer and Kan, as then $\pi_0$ of that simplicially enriched category, coincides with the following.

\hypertarget{for_categories_with_weak_equivalences}{}\subsubsection*{{For categories with weak equivalences}}\label{for_categories_with_weak_equivalences}

Given a [[category with weak equivalences]] (such as a [[model category]]), its \textbf{homotopy category} $Ho(C)$ is -- if it exists -- the [[category]] which is universal with the property that there is a [[functor]]

\begin{displaymath}
Q :  C \to Ho(C)
\end{displaymath}
that sends every weak equivalence in $C$ to an [[isomorphism]] in $Ho(C)$.

One also writes $Ho(C) := W^{-1}C$ or $C[W^{-1}]$ and calls it the [[localization]] of $C$ at the collection $W$ of weak equivalences.

More in detail, the universality of $Ho(C)$ means the following:

\begin{itemize}%
\item for any (possibly [[large category|large]]) category $A$ and functor $F : C \to A$ such that $F$ sends all $w \in W$ to isomorphisms in $A$, there exists a functor $F_Q : Ho(C) \to A$ and a natural isomorphism

\end{itemize}
\begin{displaymath}
\itexarray{
     C &&\stackrel{F}{\to}& A
     \\
     \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q}
     \\
     Ho(C)
  }
\end{displaymath}
\begin{itemize}%
\item the functor $Q^* : Func(Ho(C),A) \to Func(C,A)$ is a [[full and faithful functor]].

\end{itemize}
The second condition implies that the functor $F_Q$ in the first condition is unique up to unique isomorphism.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item If it exists, the homotopy category $Ho(C)$ is unique up to [[equivalence of categories]].


\item As described at [[localization]], in general, the morphisms of $Ho(C)$ must be constructed using zigzags of morphisms in $C$ in which the backwards-pointing arrows are weak equivalences. This means that in general, $Ho(C)$ need not be [[locally small category|locally small]] even if $C$ is. However, in many cases (such as any [[model category]]) there is a more direct description of the morphisms in $Ho(C)$ as [[homotopy]] classes of maps in $C$ between suitably ``good'' (fibrant and cofibrant) objects.


\item In [[2-limit|2-categorical terms]], the homotopy category $Ho(C)$ is the \emph{coinverter} of the canonical 2-cell

\begin{displaymath}
\itexarray{& \to \\ W & \Downarrow & C\\ & \to}
\end{displaymath}
where $W$ is the category whose objects are morphisms in $W$ and whose morphisms are commutative squares in $C$.



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

\begin{itemize}%
\item In classical [[homotopy theory]], \emph{the} homotopy category refers to the homotopy category [[Ho(Top)]] of [[Top]] with weak equivalences taken to be [[weak homotopy equivalences]].


\item [[Ho(Top)]] is often restricted to the [[full subcategory]] of spaces of the [[homotopy type]] of a [[CW-complex]] (the full subcategory of CW-complexes in $Ho(Top)$). This is equivalent to $Ho(sSet_{Quillen})$, the homotopy category of the standard Quillen-[[model structure on simplicial sets]]. This equivalence is one aspect of the [[homotopy hypothesis]].


\item In [[homological algebra]] the localization of the [[category of chain complexes]] at the [[quasi-isomorphisms]] is called the \emph{[[derived category]]}. But see also at \emph{[[homotopy category of chain complexes]]}.


\item In [[stable homotopy theory]] one considers the [[stable homotopy category]] of [[spectra]].


\item In [[equivariant stable homotopy theory]] one considers the [[equivariant stable homotopy category]] of spectra.


\item For the homotopy category of [[Cat]], see [[Ho(Cat)]].


\item For the homotopy category of that of [[combinatorial model categories]] see [[Ho(CombModCat)]].



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

\begin{itemize}%
\item [[homotopy category of a model category]]


\item [[homotopy category of an (infinity,1)-category]]


\item [[homotopy 2-category]]



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

See the references at [[model category]].

[[!redirects homotopy categories]]



\end{document}