From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10331 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: =?UTF-8?Q?Andr=c3=a9e_Ehresmann?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Discrete fibrations vs. functors into Set Date: Thu, 3 Dec 2020 12:10:55 +0100 Message-ID: References: Reply-To: =?UTF-8?Q?Andr=c3=a9e_Ehresmann?= Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8"; format=flowed Content-Transfer-Encoding: 8bit Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="16316"; mail-complaints-to="usenet@ciao.gmane.io" Cc: Categories To: Uwe Egbert Wolter Original-X-From: majordomo@rr.mta.ca Fri Dec 04 03:54:49 2020 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.74]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1kl1FA-00048I-54 for gsmc-categories@m.gmane-mx.org; Fri, 04 Dec 2020 03:54:48 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:44522) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1kl1A2-0002Pb-Dh; Thu, 03 Dec 2020 22:49:30 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1kl1DE-0003BR-8T for categories-list@rr.mta.ca; Thu, 03 Dec 2020 22:52:48 -0400 In-Reply-To: Content-Language: en-GB Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10331 Archived-At: Answer to Uwe, The notion of a discrete fibration, and its equivalence with a functor to Sets as well as with the action of a category on a set,?? were initially introduced by Charles Ehresmann in "Gattungen von Lokalen Strukturen" (Jahresbericht d. DMV Bd. 60 (1957) S. 4 9 ??? 7 7, reprinted in https://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_II_1.pdf The first category you describe is generally called the category of diagrams into Sets, Diag(Sets); the second one which is isomorphic, is called the category of (morphisms between) discrete fibrations. The category Diag(Sets), and more generally the (2-)category Diag(H) for any category H, have been extensively studied by Ren?? Guitart in some 1970's papers, in particular in D??compositions et lax-compl??tions, (avec L. Van den Bril), CTGD XVIII,4, p. 333-407, 1977. http://archive.numdam.org/article/CTGDC_1977__18_4_333_0.pdf In the last years, with Alexandre Popoff, C. Agon and M. Andreatta, we have studied and applied Diag(H) in papers on Math/Music theory, naming its objects "Poly-Klumpenhouwer-Nets" (or PK-Net) with values in H, for instance in "From Nets to PK-Nets: a categorical approach", /Perspective of new music /54-2, 2016, 5-68. For other. references, consult my personal site https://ehres.pagesperso-orange.fr/ Kind regards Andr??e [For admin and other information see: http://www.mta.ca/~cat-dist/ ]