From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9180 Path: news.gmane.org!.POSTED!not-for-mail From: Ronnie Newsgroups: gmane.science.mathematics.categories Subject: Re: About the cartesian closedness of the category of all small diagrams Date: Thu, 13 Apr 2017 20:00:00 +0000 Message-ID: Reply-To: Ronnie NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1492180545 14506 195.159.176.226 (14 Apr 2017 14:35:45 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 14 Apr 2017 14:35:45 +0000 (UTC) To: gaucher , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Apr 14 16:35:40 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1cz2K4-0003fJ-Nz for gsmc-categories@m.gmane.org; Fri, 14 Apr 2017 16:35:40 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:40934) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1cz2JH-00038d-E3; Fri, 14 Apr 2017 11:34:51 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1cz2Ic-0003JQ-2y for categories-list@mlist.mta.ca; Fri, 14 Apr 2017 11:34:10 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9180 Archived-At: This is something of an answer to question 2). I was very influenced by Streicher, T. =E2=80=98Fibred categories `a la B=C2=B4enabou=E2=80=99. http://www.mathematik.tu-darmstadt.de/=E2=88=BCstreicher/ (1999) 1=E2=80= =9385. to see how useful fibrations and cofibrations of categories for giving=20 the abstract background to constructions that occurred commonly in my=20 work with Higgins and Loday on nonabelian colimit constructions in =20 homotopy theory. So Sivera and I put these in Appendix B of the book=20 "Nonabelian Algebraic Topology" (EMS 2011) (NAT). Note that Higgins=20 pioneered in effect the use of the functor Ob: Groupoids \to Sets as a=20 cofibration of categories, (see his Notes on Categroies and Groupoids"=20 TAC Reprint) and Section B.3 of NAT gives suitable general results as=20 background to say colimits of groupoids, knowing them for sets. There is more sophisticated material in the above lectures which I have=20 not managed to use. Any advice on this could be useful! Ronnie Brown ------ Original Message ------ From: "gaucher" To: categories@mta.ca Sent: 13/04/2017 15:13:46 Subject: categories: About the cartesian closedness of the category of=20 all small diagrams >Dear categorists, > >I have three questions, the first one is a mathematical question, the >second one a bibliographical question and the last one is a speculative >question. > >1) Let K be a complete, cocomplete and cartesian closed category. >Consider the category DK of all small diagrams over K. The objects are >all small diagrams F:I-->K from a small category I to K. And a map from >(F:I-->K) to (G:J-->K) is a functor f:I-->J together with a natural >transformation mu:F-->Gf. DK is complete and cocomplete and I would=20 >like >to know if it is cartesian closed as well. > >2) My question was initially posted in >https://mathoverflow.net/q/266597/24563. From MathOverflow, I now know >that the functor DK-->Cat forgetting K is a fibred category. Since=20 >then, >I browsed the Borceux book's chapter devoted to fibred categories=20 >(Vol.2 >Chap.8). Is there other reference you could recommend me ? > >3) I also would like to know what is known about the link between >locally presentability and fibred category. Googling these terms or >looking them up in MathSciNet together gives nothing relevant.=20 >Actually, >my motivation is to know whether D(DeltaTop) and D(SimplicialSet) are >locally presentable and cartesian closed (DeltaTop is the category of >Delta-generated spaces, and SimplicialSet the category of simplicial >sets). Therefore I would like to conclude this email with a speculative >question: is there a general philosophy to deduce from the properties=20 >of >the fibers of a fibred category E-->B the same property on E ? In the >case of DK-->Cat, the fiber over I is the well-known category of >I-shaped diagrams over K... > > > >Philippe Gaucher. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]