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*To*: categories@mta.ca*From*: Steve Lack <stevel@maths.usyd.edu.au>*Date*: Fri, 28 Jul 2000 14:31:10 +1000 (EST)*In-Reply-To*: <200007241456.JAA15586@fermat.math.luc.edu>*References*: <200007241456.JAA15586@fermat.math.luc.edu>*Sender*: cat-dist@mta.ca

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 >

**References**:**No Subject***From:*"Todd H. Trimble" <trimble@math.luc.edu>

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