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# generalized van Kampen-Siefert theorem for double groupoids

# 1 The Generalized van Kampen-Siefert Theorem for Double Groupoids (GDvKST)

This theorem was first published by Ronald Brown and coworkers; please see the cited references. The following presentation of the GDvKST follows closely Dr. Ronald Brown’s presentation.

# 1.1 Double groupoids and connections

Suppose we are given a cover $\mathcal{U}$ of $X$. Then the homotopy double groupoids in the following $\rho$-sequence of the cover are well-defined:

$\bigsqcup_{{(U,V)\in\mathcal{U}\,^{2}}}\hdgb(U\cap V)\;\overset{a}{\underset{b% }{\rightrightarrows}}\bigsqcup_{{U\in\mathcal{U}}}\hdgb(U)\labto{c}\hdgb(X).$ | (1.1) |

The morphisms $a,b$ are determined by the inclusions

$a_{{UV}}:U\cap V\rightarrow U,b_{{UV}}:U\cap V\rightarrow V$ |

for each $(U,V)\in\mathcal{U}\,^{2}$ and $c$ is determined by the inclusion $c_{U}:U\rightarrow X$ for each $U\in\mathcal{U}$.

The next section presents without proof the generalization of the van Kampen-Siefert theorem for double groupoids that was first proven by Professor Ronald Brown et al; for further details the reader is referred to the original articles listed in the following Bibliography.

# 1.2 GDvKS Theorem for Double Groupoids

The following is a statement of the Generalized van Kampen Theorem (GvKT) expressed in terms of Double Groupoids with connections as developed and proven in ref. [24].

[Generalized van Kampen theorem for double groupoids]

If the interiors of the sets of $\mathcal{U}$ cover $X$, then in the above $\rho$-sequence of the cover, $c$ is the coequaliser of $a,b$ in the category of double groupoids with connections.

A special case of this result is when $\mathcal{U}$ has two elements. In this case the coequaliser reduces to a pushout.

Proof. The reader is referred to Brown et al.(2004a), [24] for the complete proof of the generalized van Kampen theorem.

Note that this theorem is a generalization of an analogous Van Kampen theorem for the fundamental group [22]. From this theorem, one can compute a particular fundamental group $\pi_{1}(X,x_{0})$ using combinatorial information on the graph 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:

(i) The computation of the topological invariant one wants to obtain the fundamental group, is obtained from the computation of a larger structure, and so part of the work is to give methods for computing the smaller structure from the larger one. This usually involves non-canonical choices, such as that of a maximal tree in a connected graph. The previous work on applying groupoids to groups gives many examples of such methods [41].

(ii) The fact that the computation can be done is surprising in two ways: (a) The fundamental group is computed 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 non-commutative invariant, which is usually even more difficult to compute precisely.

The reason for this success seems to be that the fundamental groupoid $\pi_{1}(X,X_{0})$ contains
information in *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 higher dimensions.

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## Mathematics Subject Classification

57N20*no label found*55Q15

*no label found*55N35

*no label found*

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