From: "Todd H. Trimble" <trimble@math.luc.edu>
To: categories@mta.ca
Subject: re: query: presheaf construction
Date: Sat, 29 Jul 2000 06:57:56 -0500 (CDT) [thread overview]
Message-ID: <200007291157.GAA26395@fermat.math.luc.edu> (raw)
> 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]
next reply other threads:[~2000-07-29 11:57 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-07-29 11:57 Todd H. Trimble [this message]
-- strict thread matches above, loose matches on Subject: below --
2000-07-24 14:56 Todd H. Trimble
2000-07-28 4:31 ` Steve Lack
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