Editing Groupoid (section)

=== Vertex groups and orbits ===
Given a groupoid ''G'', the '''vertex groups''' or '''isotropy groups''' or '''object groups''' in ''G'' are the subsets of the form ''G''(''x'',''x''), where ''x'' is any object of ''G''. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

The '''orbit''' of a groupoid ''G'' at a point <math>x \in X</math> is given by the set <math>s(t^{-1}(x)) \subseteq X</math> containing every point that can be joined to x by a morphism in G. If two points <math>x</math> and <math>y</math> are in the same orbits, their vertex groups <math>G(x)</math> and <math>G(y)</math> are [[group isomorphism|isomorphic]]: if <math>f</math> is any morphism from <math>x</math> to {{tmath|1= y }}, then the isomorphism is given by the mapping {{tmath|1= g\to fgf^{-1} }}.

Orbits form a partition of the set X, and a groupoid is called '''transitive''' if it has only one orbit (equivalently, if it is [[connected (category theory)|connected]] as a category). In that case, all the vertex groups are isomorphic (on the other hand, this is not a sufficient condition for transitivity; see the section [[Groupoid#Examples|below]] for counterexamples).