categories: re: query: presheaf construction

["Todd H. Trimble" <trimble@math.luc.edu>]

> 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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