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\begin{document}

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\section*{homotopy type}

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

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

[[!include homotopy - contents]]

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{homotopy_types}{Homotopy $n$-types}\dotfill \pageref*{homotopy_types} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_topological_spaces}{In topological spaces}\dotfill \pageref*{in_topological_spaces} \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}

Traditionally, a \emph{homotopy type} is a [[topological space]] regarded up to [[weak homotopy equivalence]], (although this may sometimes be referred to as its \emph{weak homotopy type}, (see below)). Formally this may be taken to mean the [[object]] that $X$ represents in the standard [[homotopy category]] [[Ho(Top)]], or, better, in the [[(∞,1)-category]] [[∞Grpd]] $\simeq$ $L_{whe} Top$, the [[simplicial localization]] of the category [[Top]] at the [[weak homotopy equivalences]], of which $Ho(Top)$ is the [[decategorification]]. As such, topological spaces regarded as homotopy types are equivalently \emph{[[∞-groupoids]]} (see at \emph{[[homotopy hypothesis]]} for more on this).

More generally, then, we may think of every object in any [[(∞,1)-topos]] $\mathcal{C}$ as being a \emph{homotopy type in the world of} $\mathcal{C}$ (just as we may think of an object of a 1-topos $\mathcal{S}$ as being a ``set in the world of $\mathcal{S}$''). For instance, if $\mathcal{C} = Sh_\infty(C)$ is an [[(∞,1)-category of (∞,1)-sheaves]]/of [[∞-stacks]] over some [[(∞,1)-site]] $C$, then an object of $\mathcal{C}$ may be thought of as a \emph{homotopy type over $C$}, or a \emph{sheaf of homotopy types}. If $\mathcal{C}$ is the [[classifying topos]] of some [[geometric theory]] $T$, then an object of $\mathcal{C}$ may be called a ``$T$-structured homotopy type''. And if $\mathcal{C}$ is a [[cohesive (∞,1)-topos]], an object of $\mathcal{C}$ may be called a ``cohesive homotopy type''. In the special case that $\mathcal{C} = Sh_\infty(*) \simeq \infty Gprd$, this reproduces the traditional notion.

The reason this makes sense is that any $(\infty,1)$-topos has an [[internal language]], which is \emph{[[homotopy type theory]]} --- a formal [[logic]] whose basic objects are abstract things called \emph{homotopy types}, just as the basic objects of [[set theory]] are abstract things called [[sets]]. Inside the internal logic of $\mathcal{C}$, its objects behave like classical homotopy types (although the ambient logic is [[constructive logic|constructive]]). This explains why we can think of objects of $\mathcal{C}$ as ``homotopy types in the world of $\mathcal{C}$'': they are the [[categorical semantics]] of these abstract homotopy types in the internal logic of $\mathcal{C}$. In the special case of $\infty Grpd$, the internal and external logic are the same, so this meaning also includes the classical usage of ``homotopy type''.

\hypertarget{homotopy_types}{}\subsubsection*{{Homotopy $n$-types}}\label{homotopy_types}

A homotopy type that is an \emph{[[n-truncated object in an (∞,1)-category]]} or equivalently that interprets a [[type]] of \emph{[[homotopy level]]} $n+2$ is also called a \emph{[[homotopy n-type]]} or \emph{$n$-type} for short. For topological spaces / [[∞-groupoids]] this means that all [[homotopy groups]] above degree $n$ are trivial.

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

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

In traditional [[homotopy theory]] of [[topological spaces]] one sometimes distinguishes the notion of \emph{strong homotopy types} from that of \emph{weak homotopy types}, depending on whether one regards topological spaces up to \emph{[[homotopy equivalence]]} or up to \emph{[[weak homotopy equivalence]]} (see also there the section \emph{\href{weak+homotopy+equivalence#RelationToHomotopyTypes}{Relation to homotopy types}}).

The two notions agree on good [[cofibrant object|cofibrant spaces]], namely on the [[CW-complexes]] (see [[model structure on topological spaces]]) and for homotopy theory proper it is the \emph{weak} homotopy equivalences that matter.

More precisely, weak homotopy equivalences between spaces give an equivalence relation on the class of topological spaces, and referring to a \emph{homotopy type} means that you are to consider properties of a space that are shared by any of the spaces weakly equivalent to it and thus in that [[equivalence class]]. In this expanded sense, therefore, a \textbf{homotopy type} is such a weak equivalence class of spaces. Using the terminology from [[homotopy category]], two spaces \emph{have the same homotopy type} if they are isomorphic in [[Ho(Top)]].

By standard theorems, homotopy types in topological can also be `modeled' by many other structures, such as

\begin{itemize}%
\item [[simplicial set|simplicial sets]] up to weak homotopy equivalence;
\item small [[category|categories]] up to weak homotopy equivalence of their [[nerves]];
\item $\infty$-[[infinity-groupoid|groupoids]] up to category-theoretic [[equivalence of categories|equivalence]] (this is the content of the [[homotopy hypothesis]]).

\end{itemize}
In most cases the tools of [[homotopy theory]], in particular [[model category|model categories]], can be used to establish these equivalences.

The sense of `modeled' is related to Whitehead's [[algebraic homotopy|algebraic homotopy theory]]. A setting such as those above acts as a model for homotopy types if there are comparison functors between $\Spaces$ and the category, and some notion of homotopy within the category yielding an equivalence of [[homotopy category|homotopy categories]].

\begin{remark}
\label{}\hypertarget{}{}
Although the notion of homotopy type is defined for arbitrary spaces, it is most usual to restrict attention to `locally [[nice topological space|nice]]' spaces such as polyhedra (i.e. realisations of [[simplicial complex|simplicial complexes]] or CW-complexes). Various other classes of space occur naturally in various parts of mathematics in particular in analysis and algebraic geometry and there the methods of abstract homotopy theory provide a way of mimicking the basic idea of homotopy type as described above.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Using variants of `weak equivalence', for instance, `$n$-equivalence', one gets coarser notions of equivalence which can be very useful. The particular case of $n$-equivalence leads to the related notion of [[homotopy n-type]].

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

\begin{itemize}%
\item [[model structure for homotopy types]]


\item [[finite homotopy type]], [[homotopy type with finite homotopy groups]]


\item [[geometric homotopy type]]


\item [[cohesive homotopy type]]


\item [[stable homotopy type]]


\item [[homotopy sphere]]



\end{itemize}
[[!include notions of type]]

[[!include homotopy n-types - table]]

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

\begin{itemize}%
\item [[Edwin Spanier]], section 7.8 of \emph{Algebraic topology}, McGraw-Hill, 1966

\end{itemize}
There are some useful points made in:

\begin{itemize}%
\item [[H. J. Baues]], \emph{[[Homotopy Types]]}, in \emph{[[Handbook of Algebraic Topology]]}, (edited by [[Ioan Mackenzie James]]), North Holland, 1995.

\end{itemize}
[[!redirects homotopy types]] [[!redirects weak homotopy type]] [[!redirects strong homotopy type]]



\end{document}