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%%% Primary Title: generalized Van Kampen theorems (HD-GVKT) -- higher dimensional
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\begin{document}
\subsection{Higher dimensional, generalized van Kampen theorems (HD-GVKT)}
There are several generalizations of the original
\htmladdnormallink{van Kampen theorem}{http://planetphysics.us/encyclopedia/VanKampenTheoremForGroupsAndGroupoids.html}, such as its
extension to crossed complexes, its extension in categorical form in terms of colimits, and its generalization to higher dimensions, i.e., its extension to \htmladdnormallink{2-groupoids}{http://planetphysics.us/encyclopedia/2Groupoid2.html}, \htmladdnormallink{2-categories}{http://planetphysics.us/encyclopedia/2Category.html} and \htmladdnormallink{double groupoids}{http://planetphysics.us/encyclopedia/WeakHomotopy.html} \cite{BHKP}.
With this HDA-GVKT approach one obtains comparatively quickly not only classical results such as the Brouwer degree and the \htmladdnormallink{relative Hurewicz theorem}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html}, but also \emph{non--commutative} results on second relative \htmladdnormallink{homotopy groups}{http://planetphysics.us/encyclopedia/ExtendedHurewiczFundamentalTheorem.html}, as well as \emph{higher dimensional} results involving the
action of, and also presentations of, the \emph{\htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/HomotopyCategory.html}}. For example,
\emph{the fundamental crossed complex} $\Pi X_*$ of the skeletal filtration of a $CW$--complex $X$ is a
useful generalization of the usual cellular chains of the universal cover of $X$. It also
gives a replacement for singular chains by taking $X$ to be the geometric realization of a singular complex of a space. Non-Abelian higher homotopy (and homology) results in \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/2Groupoid2.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html}) were proven by Ronald Brown that generalize the original van Kampen's \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} for fundamental groups (ordinary \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html}, \cite{kampen1-1933}) to \htmladdnormallink{fundamental groupoids}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} (\cite{BR67}) double groupoids, and \htmladdnormallink{higher homotopy}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} (\cite{BR-HPJ-SR2k5}); please see also Ronald Brown's presentation of the original van Kampen's theorem at PlanetMath.org \cite{VanKampen-sTheorem}.
Related research areas are: \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}, higher dimensional algebra (HDA) , higher dimensional homotopy, \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} \htmladdnormallink{homology theory}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}, \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html}, axiomatic theory of supercategories, \htmladdnormallink{n-categories}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html}, lextensive \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html}, topoi/toposes, double groupoids, omega-groupoids, crossed complexes of \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}, \htmladdnormallink{double categories}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html}, \htmladdnormallink{double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html},
\htmladdnormallink{categorical ontology}{http://planetphysics.us/encyclopedia/CategoricalOntology.html}, axiomatic foundations of Mathematics, and so on.
Its potential for applications in \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} (\htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}), and especially in \htmladdnormallink{Non-Abelian Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} (NAQAT) related to QFT, \htmladdnormallink{HQFT}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}, \htmladdnormallink{TQFT}{http://planetphysics.us/encyclopedia/SUSY2.html}, \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} and \htmladdnormallink{supergravity}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html} (\htmladdnormallink{quantum field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}) theories has also been recently pointed out and explored (\cite{BGB2k7b, BBGG1, Bgb2}).
\subsection{Generalized van Kampen theorem (GvKT)}
Consideration of a set of base points leads next to the following theorem for \emph{the fundamental groupoid}.
\subsubsection{The van Kampen theorem for the fundamental groupoid, $\pi_1(X,X_0)$, \cite{BR67}}
\emph{Let the space $X$ be the \htmladdnormallink{union}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} of open sets $U,V$ with intersection $W$, and let $X_0$
be a subset of $X$ meeting each path component of $U,V,W$. Then:}
\begin{itemize}
\item (C) (connectivity) {\em $X _0$ meets each path component of $X$, and}
\item (I) (\htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}) {\em the \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of groupoid \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} induced by inclusions:}
\end{itemize}
$$\begin{xy}
*!C\xybox{
\xymatrix{{\pi_1(W,X_0)}\ar [r]^{\pi_1(i)}\ar[d]_{\pi_1(j)}
&\pi_1(U,X_0)\ar[d]^{\pi_1(l)} \\
{\pi_1(V,X_0)}\ar [r]_{\pi_1(k)}& {\pi_1(X,X_0)} } }\end{xy}$$
\emph{is a \htmladdnormallink{pushout}{http://planetphysics.us/encyclopedia/Pushout.html} of groupoids}
\subsubsection{Remarks}
When extended to the context of double groupoids this theorem leads to a higher dimensional
generalization of the \htmladdnormallink{Van Kampen theorem}{http://planetphysics.us/encyclopedia/VanKampenTheorems.html}, the \htmladdnormallink{HD-GVKT}{http://fs512.fshn.uiuc.edu/QAT.pdf},
\cite{BHKP}.
Note that this theorem is a generalization of an analogous Van Kampen theorem for the
fundamental group, \cite{BR67, kampen1-1933}. From this theorem, one can compute a particular fundamental group $\pi_1(X,x_0)$ using combinatorial information on the \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Cod.html} of intersections of path
components of $U,V,W$, but for this it is useful to develop the algebra of groupoids. Notice
two special features of this result:
\begin{itemize}
\item (i) The \htmladdnormallink{computation}{http://planetphysics.us/encyclopedia/LQG2.html} of the \emph{invariant} one wants to obtain,
\emph{the fundamental group}, is obtained from the computation of a larger structure, and so part of the
\htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} is to give \emph{methods for computing the smaller structure from the larger one}. This
usually involves non canonical choices, such as that of a maximal \htmladdnormallink{tree}{http://planetphysics.us/encyclopedia/Tree.html} in a connected graph.
The work on applying groupoids to \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} gives many examples of such methods
\cite{HPJ2k5, BR-HPJ-SR2k5}.
\item (ii) The fact that the computation can be done at all is surprising in two ways:
(a) The fundamental group is computed {\it precisely}, even though the information for it uses input in two
dimensions, namely 0 and 1. This is contrary to the experience in homological algebra and algebraic topology, where the interaction of several dimensions involves exact sequences or spectral sequences, which give information only up to extension, and (b) the result is a \emph{non commutative invariant}, which is usually even more difficult to
compute precisely.
\end{itemize}
\subsubsection{Essential data from ref. \cite{BHKP}}
The reason for this success seems to be that the fundamental groupoid $\pi_1(X,X_0)$ contains
information in \emph{dimensions 0 and 1}, and therefore it can adequately reflect the geometry
of the intersections of the path components of $U,V,W$ and the morphisms induced by the
inclusions of $W$ in $U$ and $V$. This fact also suggested the question of whether such
methods could be extended successfully to \emph{higher dimensions}.
\begin{thebibliography}{9}
\bibitem{BR67}
R. Brown, Groupoids and Van Kampen's theorem., {\em Proc. London Math. Soc.} (3) 17 (1967) 385-401.
\bibitem{RBROWN2k6}
R. Brown, {\em Topology and Groupoids.}, Booksurge PLC (2006).
\bibitem{BHKP}
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff
space, {\em Theory and Applications of Categories.} \textbf{10} (2002) 71-93.
\bibitem{BR-AR71-2k5}
R. Brown and A. Razak, A Van Kampen theorem for unions of non-connected spaces, {\em Archiv. Math.} \textbf{42} (1984) 85-88.
\bibitem{brownjan:vkt}
R.~Brown and G.~Janelidze.:1997, {\em Van {K}ampen theorems for categories of covering morphisms in lextensive categories\/}, \emph{J. Pure Appl. Algebra}, \textbf{119}: 255--263, ISSN 0022-4049.
\bibitem{HPJ2k5}
P.J. Higgins, {\em Categories and Groupoids}, van Nostrand: New York, 1971; also {\em Reprints of Theory and Applications of Categories}, No. 7 (2005) pp 1-195.
\bibitem{BR-HPJ-SR2k5}
Brown R., Higgins P.J., Sivera, R. (2008), Non-Abelian algebraic topology, (in preparation).,
\htmladdnormallink{available here as a PDF}{http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf};
\htmladdnormallink{PDFs of other relevant HDA papers }{http://www.bangor.ac.uk/~mas010/nonab-a-t.html}.
\bibitem{VanKampen-sTheorem}
R. Brown: \htmladdnormallink{VanKampen-sTheorem}{http://planetmath.org/encyclopedia/VanKampensTheorem.html}
\bibitem{BGB2k7b}
Brown, R., Glazebrook, J. F. and I.C. Baianu.(2007), A Conceptual, Categorical and Higher Dimensional Algebra Framework of Universal Ontology and the Theory of Levels for Highly Complex Structures and Dynamics., \emph{Axiomathes} (17): 321--379.
\bibitem{kampen1-1933}
van Kampen, E. H. (1933), On the Connection Between the Fundamental
Groups of some Related Spaces, \emph{Amer. J. Math.} \textbf{55}: 261--267.
\bibitem{BBGG1}
Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook.(2006), Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L{}ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., \emph{Axiomathes}, \textbf{16} Nos. 1--2: 65--122.
\bibitem{Bggb4}
Baianu, I.C., R. Brown and J. F. Glazebrook.(2007), A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, {\em Axiomathes}, \textbf{17}: 169-225.
\bibitem{Bgb2}
Baianu, I. C., Brown, R. and J. F. Glazebrook.(2008), Quantum Algebraic Topology and Field Theories., pp.145,
\htmladdnormallink{the Monograph's PDF is here available}{http://aux.planetmath.org/files/papers/410/ANAQAT20b.pdf}(\em Preprint).
\end{thebibliography}
\end{document}
