A (finite) double groupoid is a groupoid object in the category of (finite) groupoids. They have been around since the 1960’s, arising originally in homotopy theory. Roughly speaking, they consist of a finite number of boxes with vertical and horizontal compositions and with groupoid compositions on the sides, satisfying some compatibility conditions. Let ( a field of characteristic zero) be the vector space with the boxes of as basis. The authors previously considered an algebra structure on coming from the vertical composition and a coalgebra structure from the horizontal composition, and studied when this produced a weak Hopf algebra (also called a quantum groupoid) [Publ. Mat. Urug. 10, 11-51 (2005; Zbl 1092.16021)]. A necessary and sufficient condition was given (called vacancy of ), and the resulting quantum groupoid was described as an Abelian bicrossed product.
In the paper under review, the coalgebra structure is modified. The resulting construction is shown to be a quantum groupoid if satisfies a condition called the filling condition. The resulting quantum groupoid does not seem to be describable as a bicrossed product, but it is semisimple and its finite-dimensional representation category is a semisimple rigid monoidal category. The authors show that several tensor categories fit their construction. One is the category of -bimodules over a separable algebra .