From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1585 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: re: query: presheaf construction Date: Fri, 28 Jul 2000 14:31:10 +1000 (EST) Message-ID: <14721.3214.251484.82809@milan.maths.usyd.edu.au> References: <200007241456.JAA15586@fermat.math.luc.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017948 31956 80.91.229.2 (29 Apr 2009 15:12:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:12:28 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Jul 28 13:21:44 2000 -0300 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id NAA22148 for categories-list; Fri, 28 Jul 2000 13:20:05 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from zent.mta.ca (zent.mta.ca [138.73.101.4]) by mailserv.mta.ca (8.9.3/8.9.3) with SMTP id BAA13769 for ; Fri, 28 Jul 2000 01:32:10 -0300 (ADT) X-Received: FROM siv.maths.usyd.edu.au BY zent.mta.ca ; Fri Jul 28 01:28:15 2000 -0300 X-Received: from smap@localhost by siv.maths.usyd.edu.au (8.8.8/5.7 to SMTP) id OAA51146; Fri, 28 Jul 2000 14:31:10 +1000 (EST) X-Received: from milan.maths.usyd.edu.au (stevel(.pmstaff;2406.2002)@milan.maths.usyd.edu.au) [129.78.69.163] by siv.maths.usyd.edu.au via smtpdoor V12.2 id 65627; Fri, 28 Jul 2000 14:31:10 +1000 X-Received: from stevel@localhost by milan.maths.usyd.edu.au (8.8.8/5.5 to SMTP) id OAA20723; Fri, 28 Jul 2000 14:31:10 +1000 (EST) In-Reply-To: <200007241456.JAA15586@fermat.math.luc.edu> X-Mailer: VM 6.71 under 21.1 (patch 7) "Biscayne" XEmacs Lucid Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 28 Original-Lines: 65 Xref: news.gmane.org gmane.science.mathematics.categories:1585 Archived-At: 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 >