From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1586 Path: news.gmane.org!not-for-mail From: "Todd H. Trimble" Newsgroups: gmane.science.mathematics.categories Subject: re: query: presheaf construction Date: Sat, 29 Jul 2000 06:57:56 -0500 (CDT) Message-ID: <200007291157.GAA26395@fermat.math.luc.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241017949 31960 80.91.229.2 (29 Apr 2009 15:12:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:12:29 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Jul 29 09:42:55 2000 -0300 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id JAA27230 for categories-list; Sat, 29 Jul 2000 09:37:19 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 29 Original-Lines: 52 Xref: news.gmane.org gmane.science.mathematics.categories:1586 Archived-At: > 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]