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How about Span?

Steve Lack.

 > 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. 
 > The reactions I got were varied and interesting. As filtered through 
 > me, here are some (abbreviated) responses: 
 > (1) "No, I don't think there are any examples except the obvious 
 >      locally posetal ones." 
 > (2) "The notion looks essentially algebraic, so I see no obstacle 
 >      in principle to producing examples; it should even be easy for 
 >      the right (2-categorically minded) people." 
 > (3) [From experts in domain theory] "Good question! Hmmmmmmmm....."  
 > (4) "It seems to me there is no reason in the world why examples 
 >      should not exist, but the techniques developed for dealing 
 >      with things like modest sets are probably not sufficient for 
 >      dealing with your question, and may be misleading here." 
 > The various responses suggest *to me* that the question may be 
 > quite interesting and quite hard. 
 > My own sense, based on playing around with the axioms on a purely 
 > formal level, is that there is probably no inconsistency in the sense 
 > that any two 2-cells with common source and target are provably equal. 
 > My only vague idea on producing an example would be to proceed as Church 
 > and Rosser did in the old days: work purely syntactically, and consider 
 > the possibility of strong normalization for terms. Perhaps one could 
 > then show that the term model is not locally posetal. 
 > Todd