From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6796 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: distributors for 2-categories Date: Tue, 19 Jul 2011 13:43:34 +0930 Message-ID: References: Reply-To: David Roberts NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1311086793 24922 80.91.229.12 (19 Jul 2011 14:46:33 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 19 Jul 2011 14:46:33 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Tue Jul 19 16:46:26 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QjBZ0-0006Ud-LZ for gsmc-categories@m.gmane.org; Tue, 19 Jul 2011 16:46:22 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:36766) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QjBWL-0002z6-W7; Tue, 19 Jul 2011 11:43:38 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QjBWL-0006D7-1x for categories-list@mlist.mta.ca; Tue, 19 Jul 2011 11:43:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6796 Archived-At: Hi all, =3D=3DBackground=3D=3D I have finally read Jean Benabou's Louvain lectures on distributors, whic= h he was kind enough to send me earlier this year. In them he describes the fo= llowing bicategories of distributors (some paraphrasing/simplification may occur,= and modernisation of terms - all errors are mine): Dist objects: categories C,D,... arrows: functors C^op x D--> Set (equiv. opfibrations H --> C^op x D) 2-arrows: natural transformations (equiv. cartesian functors over C^op x= D) Dist(V), for V a symmetric closed monoidal category objects: V-enriched categories C,D,... arrows: V-functors C^op x D--> V 2-arrows: V-natural transformations Dist(E), for E a regular category objects: categories internal to E C,D,... arrows: internal opfibrations H --> C^op x D 2-arrows: internal (cartesian) functors over C^op x D Dist(K), for K a bicategory objects: monads in K . . . . And here it seems to me the pattern breaks down, as taking as input the bicategories Cat, V-Cat and Cat(E), we do not arrive at any of the exampl= es on the previous list. I understand the motivation behind Dist(K), namely that one considers the process E |--> Span(E) |--> Dist(Span(E)) =3D Dist(E) (E regular category) as Cat(E) =3D Monads(Span(E)). I'm not worried about that too much. =3D=3DQuestion=3D=3D 1) Has anyone done any work on distributor-like constructions for bicateg= ories that recover the processes Cat |--> Dist, V-Cat |--> Dist(V), Cat(E) |--> Dist(E)? Something like universally adding adjoints to all 1-arrows in a bicategor= y, I would imagine. I'm asking this in the context of the equivalence between representable distributors and anafunctors, so I suppose a secondary question is: 2) Given 1) above, is there a notion of 'representable 1-arrow' in this universal construction? Thanks, David ------------------------------ David Roberts david.roberts@adelaide.edu.au University of Adelaide [For admin and other information see: http://www.mta.ca/~cat-dist/ ]