From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7673 Path: news.gmane.org!not-for-mail From: "M. Bjerrum" Newsgroups: gmane.science.mathematics.categories Subject: Re: Name for a concept? Date: 26 Apr 2013 15:07:13 +0100 Message-ID: References: <838BC420-E6C8-49A1-8AD8-5A5C45E0D496@math.ksu.edu> Reply-To: "M. Bjerrum" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1367067482 10769 80.91.229.3 (27 Apr 2013 12:58:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 27 Apr 2013 12:58:02 +0000 (UTC) Cc: David Yetter , categories To: Eduardo Pareja-Tobes Original-X-From: majordomo@mlist.mta.ca Sat Apr 27 14:58:04 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 1UW4hU-0008Cg-AW for gsmc-categories@m.gmane.org; Sat, 27 Apr 2013 14:58:00 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:38561) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UW4g6-0001Gh-TK; Sat, 27 Apr 2013 09:56:34 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UW4g6-000829-9K for categories-list@mlist.mta.ca; Sat, 27 Apr 2013 09:56:34 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7673 Archived-At: Sorry, I seem to have confused the names pseudo-protofiltered and=20 protofiltered in the previous. I took "pseudo-protofiltered" as a suggested= =20 name for connected and "span-directed", instead of just span-directed=20 (having V-cocones). But if you just ignore the "pseudo", this should not=20 disturb the content of my previous mail... i.e correction of previous, if= =20 D-filtered means "commuting with D-limits" then: 2) If D is pullbacks (V-limits) then D-filtered=3Dpseudofiltered (and not= =20 just pseudo-protofiltered) 3) If D is pullbacks and terminal objects=20 ({V,=C3=98}-limits) then D-filtered=3Dfiltered (and not just protofiltered) ..sorry for the confusion.. On Apr 26 2013, Eduardo Pareja-Tobes wrote: >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 mor= e >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 i= t >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=20 > D-filtered =3D protofiltered, or at least something similar. Something li= ke=20 > this is in p30 of [Sur la commutation des limites - Foltz] , which I got= =20 > from this MathOverflow answer:=20 > http://mathoverflow.net/questions/93262/which-colimits-commute-with-which= -limits-in-the-category-of-sets=20 > . > >So, maybe what we need to take for obtaining D-filtered =3D >span-directed/pseudo-protofiltered is D =3D coreflexive equalizers. As fin= ite >products and coreflexive equalizers =3D finite limits in SET, this would m= ean >that sifted + span-directed/pseudo-protofiltered =3D> filtered, thus >providing an affirmative answer to "It remains an open problem to determin= e >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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]