×

The compact closed bicategory of left adjoints. (English) Zbl 0980.18002

For each bicategory \({\mathcal B}\) there is a bicategory \(Sq{\mathcal B}\) of squares in \({\mathcal B}\): the objects are morphisms of \({\mathcal B}\) and the morphisms are squares in \({\mathcal B}\) containing a 2-cell. If \({\mathcal B}\) is monoidal then \(Sq{\mathcal B}\) becomes monoidal. The main result of the paper is that an object \(f\) in \(Sq{\mathcal B}\) has a right bidual if and only if \(f\) has a right adjoint and the domain and codomain of \(f\) have right duals. This implies that, if \({\mathcal B}\) is right autonomous (that is, admits all right duals), then the full sub-bicategory of \(Sq{\mathcal B}\) consisting of the left adjoint morphisms is also right autonomous.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18A10 Graphs, diagram schemes, precategories
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI