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\section*{weak homotopy equivalence}

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

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

[[!include homotopy - contents]]

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - contents]]

\hypertarget{weak_homotopy_equivalences}{}\section*{{Weak homotopy equivalences}}\label{weak_homotopy_equivalences}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_for_topological_spaces_and_simplicial_sets}{Definition (for topological spaces and simplicial sets)}\dotfill \pageref*{definition_for_topological_spaces_and_simplicial_sets} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{EquivalentCharacterizations}{Equivalent characterizations}\dotfill \pageref*{EquivalentCharacterizations} \linebreak
\noindent\hyperlink{RelationToHomotopyEquivalences}{Relation to homotopy equivalences}\dotfill \pageref*{RelationToHomotopyEquivalences} \linebreak
\noindent\hyperlink{RelationToHomotopyTypes}{Relation to homotopy types}\dotfill \pageref*{RelationToHomotopyTypes} \linebreak
\noindent\hyperlink{for_other_kinds_of_spaces}{For other kinds of spaces}\dotfill \pageref*{for_other_kinds_of_spaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ExamplesOfNonReversibleWHEs}{Of non-reversible weak homotopy equivalences}\dotfill \pageref*{ExamplesOfNonReversibleWHEs} \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}

A \emph{[[weak homotopy equivalence]]} is a map between [[topological spaces]] or [[simplicial sets]] or similar which induces [[isomorphisms]] on all [[homotopy groups]]. (The analogous concept in [[homological algebra]] is called a \emph{[[quasi-isomorphism]]}.)

The [[localization]] or [[simplicial localization]] of the categories [[Top]] and [[sSet]] at the weak homotopy equivalences used as [[weak equivalences]] yields the standard [[homotopy category]] [[Ho(Top)]] and [[Ho(sSet)]] or the [[(∞,1)-category]] of [[∞-groupoids]]/[[homotopy types]], respectively.

Weak homotopy equivalences are named after \emph{[[homotopy equivalences]]}. They can be identified with homotopy equivalences after one allows to replace the [[domains]] by a [[resolution]]. The corresponding notions in [[homological algebra]] are [[quasi-isomorphisms]] and [[chain homotopy]]-equivalences.

From another perspective, the notion of \emph{weak homotopy equivalence} is `observational', in that a map is a weak homotopy equivalence if when we look at it through the observations that we can make of it using [[homotopy groups]] or even the [[fundamental infinity-groupoid]], it looks like an equivalence. In contrast, \emph{[[homotopy equivalence]]} is more `constructive'; in that $f$ is a homotopy equivalence if there exists an inverse for it (up to homotopy, of course). Note that both of these notions are weaker than mere [[isomorphism]] of topological spaces (homeomorphism) and so can be considered examples of [[weak equivalence]]s.

There are actually two related concepts here: whether two spaces are weakly homotopy equivalent and whether a map between spaces is a weak homotopy equivalence. The former is usually defined in terms of the latter.

\hypertarget{definition_for_topological_spaces_and_simplicial_sets}{}\subsection*{{Definition (for topological spaces and simplicial sets)}}\label{definition_for_topological_spaces_and_simplicial_sets}

\begin{defn}
\label{WeakHomotopyEquivalence}\hypertarget{WeakHomotopyEquivalence}{}
For $X, Y \in$ [[Top]] or $\in$ [[sSet]] two [[topological spaces]] or [[simplicial sets]], a [[continuous function]] or [[simplicial set|simplicial map]] $f : X \to Y$ between them is called a \textbf{weak homotopy equivalence} if

\begin{enumerate}%
\item $f$ induces an [[isomorphism]] of [[connected components]] (path components in the case of topological spaces)

\begin{displaymath}
\Pi_0(f) \colon \Pi_0(X) \stackrel{\simeq}{\to} \Pi_0(Y)
\end{displaymath}
in [[Set]];


\item for all all [[points]] $x \in X$ and for all $(1 \leq n) \in \mathbb{N}$ $f$ induces an isomorphism on [[homotopy groups]]

\begin{displaymath}
\pi_n(f,x) \colon \pi_n(X,x) \stackrel{\simeq}{\to} \pi_n(Y,f(x))
\end{displaymath}
in [[Grp]].



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If $X$ and $Y$ are [[connected space|path-connected]], then (1) is trivial, and it suffices to require (2) for a single (arbitrary) $x$, but in general one must require it for at least one $x$ in each path [[connected component]].

\end{remark}
$\backslash$begin\{remark\} There are many alternative definitions of weak homotopy equivalences. A simplicial map $f$ is a weak equivalence of simplicial sets if and only if $Ex^\infty f$ is a simplicial homotopy equivalence if and only if $Hom(f,A)$ is a simplicial homotopy equivalence for any [[Kan complex]] $A$ if and only if $f$ has a right relative-homotopy-lifting property with respect to the maps $\partial\Delta^n\to\Delta^n$ if and only if $f$ is a composition of an acyclic cofibration (i.e., a map with a left lifting property with respect to all maps with a right lifting property with respect to horn inclusions) and an acyclic fibration (i.e., a map with a right lifting property with respect to inclusions $\partial\Delta^n\to\Delta^n)$. A continuous map $f$ is a weak equivalence of topological spaces if and only if $|Sing(f)|$ is a homotopy equivalence of topological spaces if and only if $Hom(A,f)$ is a homotopy equivalence for any [[CW-complex]] $A$ if and only if $f$ has a right relative-homotopy-lifting property with respect to the maps $S^{n-1}\to D^n$ if and only if $f$ is a composition of an acyclic Serre cofibration (a retract of a relative CW-complex) and an acyclic [[Serre fibration]]. Both functors $|-|$ and $Sing$ preserve and reflect weak equivalences, so any of the two classes defines the other. $\backslash$end\{remark\}

\begin{defn}
\label{}\hypertarget{}{}
The [[homotopy category]] of [[Top]] with respect to weak homotopy equivalences is [[Ho(Top)]]${}_{whe}$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Accordingly, weak homotopy equivalences are the [[weak equivalences]] in the standard [[Quillen model structure on topological spaces]] and the [[Quillen model structure on simplicial sets]], and also in the [[mixed model structure]].

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

\hypertarget{EquivalentCharacterizations}{}\subsubsection*{{Equivalent characterizations}}\label{EquivalentCharacterizations}

\begin{prop}
\label{}\hypertarget{}{}
A continuous map $f : X \to Y$ is a weak homotopy equivalence precisely if for all $n \in \mathbb{N}$ and for all [[commuting diagrams]] of continuous maps of the form

\begin{displaymath}
\itexarray{
    S^{n-1} &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    D^n &\to& Y
  }
  \,,
\end{displaymath}
where the left morphism is the inclusion of the $(n-1)$-[[sphere]] as the [[boundary]] of the $n$-[[ball]], there exists a continuous map $\sigma : D^n \to X$ that makes the resulting upper triangle commute and such that the lower triangle commutes up to a [[homotopy]]

\begin{displaymath}
\itexarray{
    S^{n-1} &&\to&& X
    \\
    \\
    \downarrow && \nearrow & \swArrow& \downarrow^{\mathrlap{f}}
    \\\\
    D^n &&\to&& Y
  }
\end{displaymath}
which is constant along $S^{n-1} \hookrightarrow D^n$.

\end{prop}
In this form the statement and its proof appears in (\hyperlink{Jardine}{Jardine}) (where it is also generalized to weak equivalences in a [[model structure on simplicial presheaves]]). See also around (\hyperlink{Lurie}{Lurie, prop. 6.5.2.1}). The relevant arguments are spelled out in (\hyperlink{May}{May, section 9.6}). A variant is called the \emph{HELP lemma} in (\hyperlink{Vogt}{Vogt}).

\hypertarget{RelationToHomotopyEquivalences}{}\subsubsection*{{Relation to homotopy equivalences}}\label{RelationToHomotopyEquivalences}

\begin{prop}
\label{}\hypertarget{}{}
Every [[homotopy equivalence]] is a weak homotopy equivalence.

\end{prop}
\begin{proof}
It requires a little bit of thought to prove this, because $f$ and its homotopy inverse $g$ need not preserve any chosen basepoint. But for any $x\in X$ and any $n\ge 1$, we have a diagram

\begin{displaymath}
\itexarray{\pi_n(X,x) & & \to & & \pi_n(X,g(f(x)))\\
  & \searrow && \nearrow && \searrow\\
  && \pi_n(Y,f(x)) && \to && \pi_n(Y,f(g(f(x))))}
\end{displaymath}
in which the two horizontal maps are isomorphisms because $g f$ and $f g$ are [[homotopy|homotopic]] to identities. Hence, by the [[two-out-of-six property]] for isomorphisms, the diagonal maps are also all isomorphisms.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Conversely, any weak homotopy equivalence between [[m-cofibrant spaces]] (spaces that are homotopy equivalent to [[CW complexes]]) is a [[homotopy equivalence]].

\end{prop}
\hypertarget{RelationToHomotopyTypes}{}\subsubsection*{{Relation to homotopy types}}\label{RelationToHomotopyTypes}

We discuss the [[equivalence relation]] generated by weak homotopy equivalence, called \emph{(weak) [[homotopy type]]}. For the ``abelianized'' analog of this situation see at [[quasi-isomorphism]] the section \emph{\href{quasi-isomorphism#RelationToChainHomologyType}{Relation to homology type}}.

\begin{prop}
\label{ReflexiveAndTransitiveButNotSymmetric}\hypertarget{ReflexiveAndTransitiveButNotSymmetric}{}
The existence of a weak homotopy equivalence from $X$ to $Y$ is a [[reflexive relation|reflexive]] and [[transitive relation]] on [[Top]], but it is not a [[symmetric relation]].

\end{prop}
\begin{proof}
Reflexivity and transitivity are trivially checked. A counterexample to symmetry is example \ref{CircleAndPseudoCircle} below.

\end{proof}
But we can consider the genuine equivalence relation \emph{generated} by weak homotopy equivalence:

\begin{defn}
\label{}\hypertarget{}{}
We say two spaces $X$ and $Y$ have the same \textbf{(weak) [[homotopy type]]} if they are equivalent under the [[equivalence relation]] \emph{generated} by weak homotopy equivalence.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Equivalently this means that $X$ and $Y$ have the same (weak) homotopy type if there exists a [[zigzag]] of weak homotopy equivalences

\begin{displaymath}
X \leftarrow \to\leftarrow \dots \to Y
  \,.
\end{displaymath}
This in turn is equivalent to saying that $X$ and $Y$ become [[isomorphism|isomorphic]] in the [[homotopy category]] [[Ho(Top)]]/[[Ho(sSet)]] with the weak homotopy equivalences [[localization|inverted]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Two spaces $X$ and $Y$ may have isomorphic homotopy groups without being weak homotopy equivalence: for this all the isomorphisms must be induced by an actual map $f : X \to Y$, as in the above definition.

However, if, roughly, one remembers, how all the homotopy groups [[nLab:action|act]] on each other, then this is enough information to exhibit the full homotopy type. This collection of data is called the \emph{[[Postnikov tower]]} decomposition of a homotopy type.

\end{remark}
\hypertarget{for_other_kinds_of_spaces}{}\subsection*{{For other kinds of spaces}}\label{for_other_kinds_of_spaces}

A map of [[simplicial sets]] is called a weak homotopy equivalence equivalently if its [[geometric realization]] is a weak homotopy equivalence of topological spaces, as above. (Since the geometric realization of any simpicial set is a [[CW complex]], in this case its geometric realization is actually a [[homotopy equivalence]].)

Likewise, a [[functor]] between [[small category|small]] categories is sometimes said to be a weak homotopy equivalence if its [[nerve]] is a weak homotopy equivalence of simplicial sets, hence of topological spaces after [[geometric realization of categories]]. These are the weak equivalences in the [[Thomason model structure]] on categories (not the [[canonical model structure]]). The statement of [[Quillen's theorem A]] and [[Quillen's theorem B]] in in this contex.

Similarly, one can define weak homotopy equivalences between any sort of object that has a [[geometric realization]], such as a [[cubical set]], a [[globular set]], an [[n-category]], an [[n-fold category]], and so on.

Note that in some of these cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of ``homotopy equivalence'' from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times \Delta^1 \to X$ and $Y\times \Delta^1 \to Y$ relating $f g$ and $g f$ to identities.

A different direction of generalization is the notion of a [[homotopy equivalence of toposes]].

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

\hypertarget{ExamplesOfNonReversibleWHEs}{}\subsubsection*{{Of non-reversible weak homotopy equivalences}}\label{ExamplesOfNonReversibleWHEs}

We discuss examples of weak homotopy equivalences that have no weak homotopy equivalence going the other way, according to prop. \ref{ReflexiveAndTransitiveButNotSymmetric} above.

\begin{example}
\label{CircleAndPseudoCircle}\hypertarget{CircleAndPseudoCircle}{}
Let $S^1 \in$ [[Top]] denote the ordinary [[circle]] and $\mathbb{S}$ the [[pseudocircle]].

There is a [[continuous function]] $S^1 \to \mathbb{S}$ which is a weak homotopy equivalence, hence in particular $\pi_1(\mathbb{S}) \simeq \mathbb{Z}$. But every continuous map the other way round has to induce the trivial map on $\pi_1$.

\end{example}
This is the simplest in a class of counter-examples discussed in (\hyperlink{McCord}{McCord}).

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

\begin{itemize}%
\item [[equality]]


\item [[isomorphism]]


\item [[equivalence]]


\item [[weak equivalence]]


\item [[homotopy equivalence]], \textbf{weak homotopy equivalence}

\begin{itemize}%
\item [[rational homotopy equivalence]]


\item [[n-equivalence]]



\end{itemize}
[[stable weak homotopy equivalence]]


\item [[Bousfield equivalence]]


\item [[homotopy equivalence of toposes]]


\item [[equivalence in an (∞,1)-category]]


\item [[equivalence of (∞,1)-categories]]



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

A general account is for instance in section 9.6 of

\begin{itemize}%
\item [[Peter May]], \emph{[[A Concise Course in Algebraic Topology]]}

\end{itemize}
The characterization of weak homotopy equivalences by lifts up to homotopy seems is in

\begin{itemize}%
\item [[Rick Jardine]], \emph{Simplicial Presheaves}, Journal of Pure and Applied Algebra 47, 1987, no.1, 35-87.

\end{itemize}
\begin{itemize}%
\item [[Rainer Vogt]], \emph{The HELP-Lemma and its converse in Quillen model categories} (\href{http://arxiv.org/abs/1004.5249}{arXiv:1004.5249})

\end{itemize}
For related and general discussion see also section 6.5 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
Examples for the non-symmetry of the weak homotopy equivalence relation are in

\begin{itemize}%
\item Michael McCord, \emph{Singular homology groups and homotopy groups of finite topological spaces}, Duke Math. J. Volume 33, Number 3 (1966), 465-474. (\href{http://projecteuclid.org/euclid.dmj/1077376525}{EUCLID})

\end{itemize}
See also

\begin{itemize}%
\item Topospaces-Wiki, \emph{\href{http://topospaces.subwiki.org/wiki/Weak_homotopy_equivalence_of_topological_spaces}{Weak homotopy equivalence of topological spaces}}

\end{itemize}
[[!redirects weak homotopy equivalences]]



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