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categories: re: query: presheaf construction
> 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.]
>> 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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