Starting form symmetric monoidal closed (= autonomous) categories, Po-Hsiang Chu showed how to construct new *-autonomous categories, i.e., autonomous categories that are self-dual by virtue of having a dualizing object. Recently, Michael Barr extended this to the nonsymmetric, but closed, case, utilizing monads and modules between them. Since these notions are well-understood for bicategories, we introduce a notion of cyclic *-autonomy for these that implies closedness and, moreover, is inherited when forming bicategories of monads and of interpolads. Since the initial step of Barr's construction also carries over to the bicategorical setting, we recover his main result as an easy corollary. Furthermore, the Chu-construction at this level may be viewed as a procedure for turning the endo-1-cells of a closed bicategory into the objects of a new closed bicategory, and hence conceptually is similar to constructing bicategories of monads and of interpolads.
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