From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3728 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: two preprints on Kan extensions in/for double categories Date: Mon, 16 Apr 2007 15:07:15 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019486 10038 80.91.229.2 (29 Apr 2009 15:38:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:38:06 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Apr 16 10:24:51 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 16 Apr 2007 10:24:51 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HdR6L-0004Ra-M5 for categories-list@mta.ca; Mon, 16 Apr 2007 10:18:37 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 15 Original-Lines: 43 Xref: news.gmane.org gmane.science.mathematics.categories:3728 Archived-At: This is to announce two preprints, which take on our study of weak double categories and deal with Kan extensions. The first two parts, dealing with limits and adjoints, respectively, where published in "Cahiers" (1999, 2004) and can also be found on my server. M. Grandis ________________ M. Grandis - R. Pare, Kan extensions in double categories (On weak double categories, Part III), Dip. Mat. Univ. Genova, Preprint 553 (2007). http://www.dima.unige.it/~grandis/Dbl3.pdf (ps) ________________ ---, Lax Kan extensions for double categories (On weak double categories, Part IV), Dip. Mat. Univ. Genova, Preprint 554 (2007). http://www.dima.unige.it/~grandis/Dbl4.pdf (ps) ________________ Abstracts. Part III. This paper deals with Kan extensions in a weak double category. Absolute Kan extensions are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category. Part IV. Right Kan extensions for weak double categories extend double limits and other constructions, called vertical companions and vertical adjoints, studied in previous papers. We prove that these particular cases are sufficient to construct all pointwise unitary lax right Kan extensions, along those lax double functors which satisfy a Conduche type condition. Double categories 'based on profunctors' are complete, i.e. have all such constructions, while the double category of commutative squares on a complete category is not, in general.