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.

Bibliography

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).