Duals invert

Richard Wood – 13 February 2008

joint work with Ross Street We show that recent theorems of Day and Pastro on Frobenius functors can be applied to give another proof of the following theorem of Walters and Wood: For A a Frobenius object in a cartesian bicategory B, for all X in B, Map(B)(X,A) is a groupoid. Suppose that an object A in a monoidal bicategory M has a right bidual A\ring. We begin by defining an exact pairing for arrows x:A--->X and y:A\ring--->X in M, for X a monoidal object. We then show that the Day/Pastro theorems extend to such pairings and Frobenius arrows f:X--->Y in M. Ultimately we apply our results in the case M=Bco, for B cartesian.