For a cover of a space , the corresponding Čech groupoid is the internal groupoid
whose set of objects is the disjoint union of the covering patches, and the set of morphisms is the disjoint union of the intersections of these patches.
This is the -coskeleton of the full Čech nerve. See there for more details.
If we speak about generalized points of the (which are often just ordinary points, in applications), then
an object of is a pair where is a point in ;
there is a unique morphism for all pairs of objects labeled by the same such that ;
hence the composition of morphism is of the form
For a smooth manifold and an atlas by coordinate charts, the Čech groupoid is a Lie groupoid which is equivalent to as a Lie groupoid:
For the Lie groupoid with one object coming from a Lie group morphisms of Lie groupoids of the form
are also called anafunctors from to . They correspond to smooth -principal bundles on .
Revised on March 2, 2012 15:56:07
by Andrew Stacey