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Tensor categories attached to double groupoids. (English) Zbl 1099.16016

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 $kT$ ($k$ a field of characteristic zero) be the vector space with the boxes of $T$ as basis. The authors previously considered an algebra structure on $kT$ 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 $T$), 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 $T$ 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 $R$-bimodules over a separable algebra $R$.

MSC:
 16W30 Hopf algebras (assoc. rings and algebras) (MSC2000) 18D05 Double categories, 2-categories, bicategories and generalizations 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories) 20L05 Groupoids