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*To*: categories@mta.ca*Subject*: categories: re: query: presheaf construction*From*: "Todd H. Trimble" <trimble@math.luc.edu>*Date*: Sat, 29 Jul 2000 06:57:56 -0500 (CDT)*Sender*: cat-dist@mta.ca

> 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. >> [rest of message snipped]

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