Bicategories of quantales
by
Jan Paseka
Department of Mathematics, Masaryk University, Janackovo nam. 2a, 662 95 Brno, Czech Republic

It is well known that rings are the objects of a bicategory, whose arrows are bimodules, composed through the bimodule tensor product. We give an analogous bicategorical description of quantales. The upshot is that known definition of Morita equivalence for this case amounts to isomorphism of objects in the pertinent bicategory.

Namely, for any two quantales A, B, let (A, B) be the collection of all bimodules A --> M <-- B, seen as the class of objects of a category, whose arrows are A-B bimodule maps. The collection of all quantales as objects, bimodules as arrows, (horizontal) composition (A, B)×(B, C) --> (A, C) given by \otimes_B, and the unit arrow 1_A in (A, A) given by the canonical bimodule A --> A <-- A, is a bicategory [Q^*]. One looks at bimodules as generalized homomorphisms. In the language of bicategories, two quantales are isomorphic objects in the bicategory [Q^*] iff they have equivalent representation categories (where the equivalence functor is required to be a SUP-functor). A characterization of invertible bimodules is given.

Date received: November 3, 2000