From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7668 Path: news.gmane.org!not-for-mail From: Eduardo Pareja-Tobes Newsgroups: gmane.science.mathematics.categories Subject: Re: Name for a concept? Date: Thu, 25 Apr 2013 20:47:49 +0200 Message-ID: References: <838BC420-E6C8-49A1-8AD8-5A5C45E0D496@math.ksu.edu> Reply-To: Eduardo Pareja-Tobes NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1366932606 12676 80.91.229.3 (25 Apr 2013 23:30:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 25 Apr 2013 23:30:06 +0000 (UTC) Cc: categories To: David Yetter Original-X-From: majordomo@mlist.mta.ca Fri Apr 26 01:30:11 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UVVc4-0003lH-3L for gsmc-categories@m.gmane.org; Fri, 26 Apr 2013 01:30:04 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:36556) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UVVat-0007VS-6B; Thu, 25 Apr 2013 20:28:51 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UVVau-000854-Ar for categories-list@mlist.mta.ca; Thu, 25 Apr 2013 20:28:52 -0300 In-Reply-To: <838BC420-E6C8-49A1-8AD8-5A5C45E0D496@math.ksu.edu> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7668 Archived-At: There is something about this, yes; I read about this sort of things some time ago, so take what follows with a grain of salt. First, I will work with the opposite of your category, so what we have is a category for which every span can be completed to a commutative square. Let's call this notion "span-directed". Obviously, this looks like some sort of generalized filteredness notion; every filtered category is span-directed. As defined, span-directed does not require connectedness, so this look more like "pseudo-filtered", which according to Mac Lane CftWM was something introduced by Verdier [SGA4I, I.2.7]. A category is pseudo-filtered iff is a coproduct of filtered categories. Now, in Definition 53 of [Protolocalisations of homological categories - Borceux, Clementino, Gran, Sousa], they name a category protofiltered if it is span-directed and connected. So, according to all this, if one was to follow the same pattern, span-directed should be named "pseudo-protofiltered" :) I think it would be possible to have a conceptual characterization of span-directed/pseudo-protofiltered categories, in terms of distributivity of colimits in SET indexed by them over some natural class of limits. That is, as D-filtered categories for D a "doctrine of D-limits" in the terminology of [A classification of accessible categories - Ad=C3=A1mek, Borceux, Lack, Rosick=C3=BD]. There, for a small class of categories D a category I is said to be D-filtered if colimits indexed by I distribute over D-limits (any diagram indexed by a category in D) in SET. Then, you get that 1. If D =3D finite categories, D-filtered =3D filtered 2. If D =3D finite connected categories, D-filtered =3D pseudo-filtered 3. If D =3D finite discrete categories, D-filtered =3D sifted Speculative content follows, possibly everything after this point is wrong: Now, I think that if you take D =3D equalizers what you get could be D-filtered =3D protofiltered, or at least something similar. Something like this is in p30 of [Sur la commutation des limites - Foltz] , which I got from this MathOverflow answer: http://mathoverflow.net/questions/93262/which-colimits-commute-with-which-l= imits-in-the-category-of-sets . So, maybe what we need to take for obtaining D-filtered =3D span-directed/pseudo-protofiltered is D =3D coreflexive equalizers. As fini= te products and coreflexive equalizers =3D finite limits in SET, this would me= an that sifted + span-directed/pseudo-protofiltered =3D> filtered, thus providing an affirmative answer to "It remains an open problem to determine whether a sifted proto=EF=AC=81ltered category is =EF=AC=81ltered": [Protolocalisations of homological categories] =E2=80=8B--=E2=80=8B =E2=80=8BEduardo Pareja-Tobes=E2=80=8B =E2=80=8Boh no sequences!=E2=80=8B On Thu, Apr 25, 2013 at 5:14 AM, David Yetter wrote: > Is there an existing name in the literature for a category in which every > cospan admits a completion to a commutative square? (Just that, no > uniqueness, no universal > properties required, just every cospan sits inside at least one > commutative square). If so, what have such things been called? If not, > does anyone have a poetic idea for a good name for > such categories? > > Best Thoughts, > David Yetter > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]