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\begin{document}

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\section*{homotopy 2-type}

\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{classification}{Classification}\dotfill \pageref*{classification} \linebreak
\noindent\hyperlink{homotopy_types_as_crossed_modules}{Homotopy $2$-types as crossed modules}\dotfill \pageref*{homotopy_types_as_crossed_modules} \linebreak
\noindent\hyperlink{homotopy_types_as_simplicial_groupoids}{Homotopy $2$-types as simplicial group(oid)s}\dotfill \pageref*{homotopy_types_as_simplicial_groupoids} \linebreak
\noindent\hyperlink{homotopy_types_as_groupoids}{Homotopy $2$-types as $2$-groupoids}\dotfill \pageref*{homotopy_types_as_groupoids} \linebreak
\noindent\hyperlink{homotopy_2types_as_double_groupoids}{Homotopy 2-types as double groupoids}\dotfill \pageref*{homotopy_2types_as_double_groupoids} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A homotopy $2$-type is a view of a space where we consider its properties only up to the $2$nd [[homotopy group]] $\pi_2$. To make this precise, we look at maps that `see' invariants in dimensions 0,1, and 2. These are the 2-equivalences:

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

A continuous map $X \to Y$ is a \textbf{homotopy $2$-equivalence} if it induces isomorphisms on $\pi_i$ for $i = 0, 1, 2$ at each basepoint. Two spaces share the same \textbf{homotopy $2$-type} if they are linked by a zig-zag chain of homotopy $2$-equivalences.

For any `nice' space $X$, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space $Y$ so that the inclusion of $X$ into $Y$ is a homotopy $2$-equivalence; up to (weak) [[homotopy equivalence]], the result is the same for any space with the same homotopy $2$-type. Accordingly, a \textbf{homotopy $2$-type} may alternatively be defined as a space with trivial $\pi_i$ for $i \gt 2$, or as the unique (weak) [[homotopy type]] of such a space, or as its fundamental $\infty$-[[fundamental infinity-groupoid|groupoid]] (which will be a $2$-[[2-groupoid|groupoid]]).

See the general discussion in [[homotopy n-type]].

\hypertarget{classification}{}\subsection*{{Classification}}\label{classification}

Homotopy $2$-types can be classified by various different types of algebraic data.

\hypertarget{homotopy_types_as_crossed_modules}{}\subsubsection*{{Homotopy $2$-types as crossed modules}}\label{homotopy_types_as_crossed_modules}

Homotopy $2$-types can be classified up to weak homotopy type by [[crossed module]]s of groupoids. These are the $2$-truncated versions of [[crossed complex]]es. Such a $C$ consists of a morphism

\begin{displaymath}
\delta: C_2 \to C_1
\end{displaymath}
of [[groupoid]]s with object set $C_0$ such that $C_2$ is totally disconnected, i.e. is a family of groups $C_2(x), x \in C_0$. Further the groupoid $C_1$ operates on this family of groups so that (using right operations) if $a: x \to y$ in $C_1$ and $u \in C_2(x)$ then $u^a \in C(y)$; and the usual rules for an operation are satisfied, namely $(uv)^a=u^a v^a$, $u^1=u$, $(u^a)^b=u^{a b}$ when these are defined. Further the two basic crossed module rules hold:

CM1) $\delta(u^a)= a^{-1} (\delta u) a$;

CM2) $v^{-1} u v = u ^{\delta v}$;

for all $a \in C_1, u,v \in C_2$ when the rules make sense.

Such a crossed module may be extended to a crossed complex $sk^2 C$ by adding trivial elements in dimensions higher than 2. Hence there is a simplicial [[nerve]] $N^\Delta C$ which in dimension $n$ is

\begin{displaymath}
Crs(\Pi (\Delta^n_*), sk^2 C).
\end{displaymath}
The geometric realisation of this is the [[classifying space]] $BC$. Its first and second homotopy groups at $x \in C_0$ are the cokernel and kernel of $\delta: C_2(x) \to C_1(x,x)$. Its components are those of the groupoid $C_1$. All other homotopy groups are trivial.

If $X$ is a CW-complex then there is a bijection of homotopy classes

\begin{displaymath}
[X,BC] \cong [\Pi X_*, sk^2 C],
\end{displaymath}
and hence there is a map $X \to B(cotr^2 \Pi X_*)$ inducing isomorphisms of homotopy groups in dimensions 1 and 2.

Here the cotruncation $cotr^n D$ of a general crossed complex $D$ agree with $D$ up to dimension $(n-1)$, is $Cok \delta_{n+1}$ in dimension $n$, and is trivial in higher dimensions.

It is in this sense that crossed modules of groupoids classify weak homotopy $2$-types.

The category $Crs^2$ of such crossed modules of groupoids is equivalent to that of strict [[2-groupoid]]s. Further, $Crs^2$ is [[closed monoidal category|monoidal closed]]:

\begin{displaymath}
Crs^2(C \otimes D, E) \cong Crs^2(C, CRS^2(D,E))
\end{displaymath}
and with a unit interval object $I$ so that (left) homotopies are determined as morphisms $Crs^2(I \otimes D,E)$ or as elements of $CRS^2(D,E)_1$.

\hypertarget{homotopy_types_as_simplicial_groupoids}{}\subsubsection*{{Homotopy $2$-types as simplicial group(oid)s}}\label{homotopy_types_as_simplicial_groupoids}

As a crossed module give rise to an internal groupoid in the category of groups (or groupoids), we can take the nerve of that structure and get a simplicial group (or [[simplicial groupoid|simplicially enriched groupoid]]). From a simplicial group(oid), $G$, one can define a simplicial set called the classifying space $\overline{W}G$ of the simplicial group, $G$, for which construction see [[simplicial group]]. We thus can start with a crossed module $C$ form a simplicial group and then take $\overline{W}$ of that to get another model of $\mathcal{B}C$.

\hypertarget{homotopy_types_as_groupoids}{}\subsubsection*{{Homotopy $2$-types as $2$-groupoids}}\label{homotopy_types_as_groupoids}

With respect to the standard [[homotopy theory]]-structure on [[2-groupoids]] ([[2-truncated]] [[infinity-groupoids]]) these are equivalent to homotopy 2-types. See at \emph{[[homotopy hypothesis]]} for more on this.

\hypertarget{homotopy_2types_as_double_groupoids}{}\subsubsection*{{Homotopy 2-types as double groupoids}}\label{homotopy_2types_as_double_groupoids}

see

\begin{itemize}%
\item [[Antonio Martínez Cegarra]]; Benjamn A. Heredia ; Josu\'e{} Remedios, \emph{Double groupoids and homotopy 2-types} Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also \href{http://arxiv.org/abs/1003.3820}{arXiv:1003.3820}.

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

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

[[!redirects 2-type]] [[!redirects 2-types]]

[[!redirects homotopy 2-types]]



\end{document}