How about Span?
Steve Lack.
Since Ladj(Span) is essentially Set, we would need, for every set b, a set pb such that for every a, the large category of spans a -|-> b is equivalent to the small discrete category of functions a --> pb. This doesn't work. [Just to avoid a possible misunderstanding: if B is a bicategory, then by Ladj(B) I mean the locally full subbicategory of B with the same objects as B and whose 1-cells are left adjoints in B. Katis and Walters have a paper which uses the same notation Ladj(B) for something else.] -- Todd.
At the Como meeting last week, I asked various people a question which I view as having foundational significance: is there a setting in which one can iterate the presheaf construction (as free cocompletion) without ever having to use the word "small" or worry about size?
Here is a more precise formulation of what I am after. I want an example of a compact closed bicategory B [think: bicategory of profunctors] with the following very strong property: the inclusion
i: Ladj(B) --> B,
of the bicategory of left adjoints in B, has a right biadjoint p such that, calling y: 1 --> pi the unit and e: ip -|-> 1 the counit, the isomorphisms which fill in the triangles iy yp i --> ipi p --> pip \ | \ | \ | ei \ | pe \| \| i p
furnish the unit and counit, respectively, of adjunctions iy --| ei in B and pe --| yp in Ladj(B). (These structures should also be compatible with the symmetric monoidal bicategory structures on B and Ladj(B).) By exploiting compact closure, it's easy to see that p(b) is equivalent to an exponential (p1)^(b^op) in Ladj(B), where b^op denotes the dual of b in the sense of compact closure. So the unit y: 1 --> pi takes the yoneda-like form b --> v^(b^op); the axioms imply it is the fully faithful unit of a KZ-monad.
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