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How about Span?
> 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.
- No Subject
- From: "Todd H. Trimble" <email@example.com>