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Hometopological groupoid

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# topological groupoid

A *groupoid* is a set $G$ together with a subset $G_{2}\subset G^{2}$ of composable pairs, a *multiplication* $\mu:G_{2}\to G:(a,b)\mapsto ab$ and an *inversion* $\cdot^{{-1}}:G\to G:a\mapsto a^{{-1}}$ such that

1. $\cdot^{{-1}}\circ\cdot^{{-1}}=\mathrm{id}_{G}$,

2. if $\{(a,b),(b,c)\}\subset G_{2}$ then $\{(ab,c),(a,bc)\}\subset G_{2}$ and $(ab)c=a(bc)$,

3. $(b,b^{{-1}})\in G_{2}$ and if $(a,b)\in G_{2}$ then $abb^{{-1}}=a$ and

4. $(b^{{-1}},b)\in G_{2}$ and if $(b,c)\in G_{2}$ then $b^{{-1}}bc=c$.

Furthermore we have the *source* or *domain map* $\sigma:G\to G:a\mapsto a^{{-1}}a$ and the *target* or *range map* $\tau:G\to G:a\mapsto aa^{{-1}}$. The image of these maps is called the *unit space* and denoted $G_{0}$. If the unit space is a singleton than we regain the notion of a group.

We also define $G_{a}:=\sigma^{{-1}}(\{a\})$, $G^{b}:=\tau^{{-1}}(\{b\})$ and $G_{a}^{b}:=G_{a}\cap G^{b}$. It is not hard to see that $G_{a}^{a}$ is a group, which is called the *isotropy group* at $a$.

We say that a groupoid $G$ is *principal* and *transitive*, if the map $(\sigma,\tau):G\to G_{0}\times G_{0}$ is injective and surjective, respectively.

A groupoid can be more abstractly and more succinctly defined as a category whose morphisms are all isomorphisms.

A *topological groupoid* is a groupoid $G$ which is also a topological space, such that the multiplication and inversion are continuous when $G_{2}$ is endowed with the induced product topology from $G^{2}$. Consequently also $\sigma$ and $\tau$ are continuous.

# References

- 1 P.J. Higgins, Categories and groupoids, van Nostrand original, 1971; Reprint Theory and Applications of Categories, 7 (2005) pp 1-195.
- 2 R. Brown, Topology and groupoids, xxv+512pp, Booksurge 2006.
- 3 R. Brown, ‘Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions’, Proceedings of the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo (Poland) 8.05.2005-15.05.2005, Banach Centre Publications 76, Institute of Mathematics Polish Academy of Sciences, Warsaw, (2007) 51-63. (math.DG/0602499).

## Mathematics Subject Classification

18B40*no label found*20L05

*no label found*54H13

*no label found*54H11

*no label found*

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## Comments

## Classification

Sorry, I didn't see your comment in the entry. What seems to be the problem? Have you tried leaving off the "msc:" part?

Cam

## Re: Classification

I finally figured it out, I forgot to separate with comma's...