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Kan Complexes can be seen as a generalization of groupoids, mostly called (weak) infinity groupoids in this context.

On groupoids we can define the \textbf{group of bisections} the following way:

Let $G \Rightarrow M$ be a groupoid with arrow set $G$, object set $M$, target $d_0: G \to M$, source $d_1: G \to M$, 'units' $s_0: M \to G$ and product
$*:G \times_{(d_0\times d_1)} G \to G$

Then a bisection is a map $\sigma: M \to G$ right inverse to $d_1: G \to M$, such that $d_0\circ b : M \to M$ is bijective.

(Of course this would work similar if we say b is right inverse to $d_0$ and $d_1\circ b$ is a bijection)

The set of bisections has a group structure, with product defined on elements of M by $$(b_1 \star b_2)(m) := b_1(d_0\circ b_2(m))*b_2(m)$$
and unit s_0 .

Now, the question is, if this can be generalized or reformulated to something on oo-groupoids, replacing the associative product by something else.

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Seems like I have taggin problems on this computer. Can someone please tag it with groupoids, higher category theroy, too? –  Nevermind Apr 17 '13 at 13:02

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