A twisted pair groupoid

h is an arrow from a to b in the groupoid, while g is an arrow from b to c. Their composition is gh. The arrow h may be freely varied, while g may only vary on the source fibre through b, in order to obtain a well-defined composition.

The red and blue lines represent the source and target fibres, respectively. The black line represents the identity arrows. The purple line represents the isotropy group at the point where the twist is performed.

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Marco Gualtieri and Brent Pym, 29 August 2013, Created with GeoGebra