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\begin{document}

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\section*{higher homotopy van Kampen theorem}

\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{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{ForTopSpaces}{For topological spaces}\dotfill \pageref*{ForTopSpaces} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{strict_version}{Strict version}\dotfill \pageref*{strict_version} \linebreak
\noindent\hyperlink{for_objects_in_a_cohesive_topos}{For objects in a cohesive $(\infty,1)$-topos}\dotfill \pageref*{for_objects_in_a_cohesive_topos} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

One form of a \emph{higher homotopy van Kampen theorem} is a theorem that asserts that the [[homotopy type]] of a [[topological space]] can be computed by a suitable [[colimit]] or [[homotopy colimit]] over homotopy types of its pieces. Another form which allows specific computation deals with spaces with certain kinds of structure, for example filtered spaces or $n$-cubes of spaces.

This generalizes the [[van Kampen theorem]], which only deals with the underlying 1-type (the fundamental groupoid).

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\hypertarget{ForTopSpaces}{}\subsubsection*{{For topological spaces}}\label{ForTopSpaces}

\hypertarget{general}{}\paragraph*{{General}}\label{general}

\begin{utheorem}
Let $X$ be a [[topological space]], write $Op(X)$ for its [[category of open subsets]] and let

\begin{displaymath}
\chi : C \to Op(X)
\end{displaymath}
be a [[functor]] out of a [[small category]] $C$ such that

\begin{itemize}%
\item for each point $x\in X$ the [[full subcategory]] $C_x$ of objects $c$ such that $\chi(x)$ contains $x$ has a [[weak homotopy equivalence|weakly]] [[contractible]] [[nerve]].

\end{itemize}
Then:

the canonical morphism in [[sSet]] out of the [[colimit]]

\begin{displaymath}
{\lim_\to} Sing \circ \chi \to Sing(X)
\end{displaymath}
into the [[singular simplicial complex]] of $X$ exhibits $Sing(X)$ as the [[homotopy colimit]] $hocolim Sing \circ \chi$.

\end{utheorem}
This is theorem A.1.1 in (\hyperlink{Lurie}{Lurie}).

\hypertarget{strict_version}{}\paragraph*{{Strict version}}\label{strict_version}

The following is a version of the above general statement restricted to a [[strict ∞-groupoid]]-version of the [[fundamental ∞-groupoid]] and applicable for [[topological spaces]] that are equipped with the extra structure of a [[filtered topological space]].

Notice that these strict $\infty$-groupoids are equivalent to [[crossed complex]]es.

Suppose $X_*$ is a [[filtered space]] and $X$ is the union of the interiors of sets $U^i$, $i \in I$. Let $U^i_*$ be the filtered space given by the intersections $U^i \cap X_n$ for $n \geq 0$. If $d=(i,j) \in I^2$ we write $U^d$ for $U^i \cap U^j$. We then have a coequaliser diagram of filtered spaces

\begin{displaymath}
\bigsqcup_{d \in I^2} U^d_* \rightrightarrows ^a_b \bigsqcup _{i \in I} U^i_* \to ^c X_*.
\end{displaymath}
\begin{utheorem}
If the filtered spaces $U^f_*$ are [[connected filtered space]]s for all finite intersections $U^f_*$ of the filtered spaces $U^i_*$, then

\begin{enumerate}%
\item (Conn) The filtered space $X_*$ is connected; and


\item (Iso) The fundamental [[crossed complex]] functor $\Pi$ takes the above coequaliser diagram of filtered spaces to a coequaliser diagram of crossed complexes.



\end{enumerate}
A full account is given in (\hyperlink{NAT}{Brown-Higgins-Sivera} and the methodology is discussed in (\hyperlink{RBrown}{Brown}).

\end{utheorem}
\textbf{Remarks}

\begin{itemize}%
\item Note that because $\Pi$ uses groupoids, it obviously takes disjoint unions $\bigsqcup$ of filtered spaces into disjoint unions (= coproducts) $\bigsqcup$ of crossed complexes.


\item The proof of the theorem is not direct but goes via the fundamental cubical $\omega$-groupoid with connections of the filtered spaces, as that context allows the notions of \emph{algebraic inverse to subdivision} and of \emph{commutative cube}. However the proof is a direct generalisation of a proof for the [[van Kampen theorem]] for the [[fundamental groupoid]].


\item Applications of this theorem include many basic facts in algebraic topology, such as the Relative Hurewicz Theorem, the Brouwer degree theorem, and new nonabelian results on 2nd relative homotopy groups, not of course obtainable by the traditional wholly abelian methods. No use is made of \emph{singular homology theory} or of \emph{simplicial approximation}. Also included is a version of the ``small simplex theorem'', see Theorem 10.4.20 of (\hyperlink{NAT}{Brown-Higgins-Sivera}).



\end{itemize}
\hypertarget{for_objects_in_a_cohesive_topos}{}\subsubsection*{{For objects in a cohesive $(\infty,1)$-topos}}\label{for_objects_in_a_cohesive_topos}

In a [[cohesive (∞,1)-topos]] (already in a [[locally ∞-connected (∞,1)-topos]]) higher van Kampen theorems hold in great generality.

See the section .

In particular for the cohesive $(\infty,1)$-topos [[?TopGrpd]] of [[topological ∞-groupoid]]s this reproduces the \hyperlink{ForTopSpaces}{topological higher van Kampen theorem} discussed above.

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

Here is one application in dimension 2 not easily obtainable by traditional [[algebraic topology]].

Let $0 \to P \to Q \to R \to 0$ be an [[exact sequence]] of [[abelian groups]]. Let $X$ be the [[mapping cone]] of the induced map $K(P,1) \to K(Q,1)$ of [[Eilenberg-Mac Lane space]]s. Then a [[crossed module]] representing the [[homotopy 2-type]] of $X$ is $\mu: C \to Q$ where $C$ is abelian and is the direct sum $\oplus_{r \in R} P^r$ of copies of $P$ one for each $r \in R$ and the action of $Q$ is via $R$ and permutes the copies by $(p,r)^s=(p,r+s)$. Similar examples for $P,Q,R$ nonabelian are do-able, more complicated, and certainly \textbf{not} obtainable by traditional methods.

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

The version for topological spaces and the [[fundamental infinity-groupoid]] functor is discussed in Appendix A of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Ek-Algebras]]}

\end{itemize}
The version for filtered topological spaces and the strict homotopy $\infty$-groupoid functor is discussed in

\begin{itemize}%
\item [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], \emph{[[Nonabelian Algebraic Topology]]} (\#NAT)

\end{itemize}
while the general methodology is discussed in

\begin{itemize}%
\item [[Ronnie Brown]] ``Modelling and computing homotopy types: I'' to appear in 2017 in a special issue of Indagationes Math in honour of L.E.J. Brouwer. (https://arxiv.org/abs/1610.07421) (\#RBrown)

\end{itemize}
One area of application of work of Brown and Loday is to a nonabelian tensor product of groups, see:

\begin{itemize}%
\item [[Ronnie Brown]] Bibliography on the nonabelian tensor product. (http://www.groupoids.org.uk/nonabtens.html) (\#tens)

\end{itemize}
[[!redirects Higher Homotopy van Kampen Theorem]] [[!redirects higher homotopy van Kampen Theorem]]

[[!redirects higher van Kampen theorem]]



\end{document}