From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7671 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 14:03:52 +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 1367067345 9690 80.91.229.3 (27 Apr 2013 12:55:45 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 27 Apr 2013 12:55:45 +0000 (UTC) Cc: David Yetter , categories To: Eduardo Pareja-Tobes Original-X-From: majordomo@mlist.mta.ca Sat Apr 27 14:55:49 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 1UW4fM-0005Qz-Sk for gsmc-categories@m.gmane.org; Sat, 27 Apr 2013 14:55:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:38549) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UW4dd-0000mg-F3; Sat, 27 Apr 2013 09:54:01 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UW4dc-0007xP-Uo for categories-list@mlist.mta.ca; Sat, 27 Apr 2013 09:54:00 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7671 Archived-At: Hello, I suppose that a span is a diagram of finitely many arrows of same domain= =20 (or the op-situation). And the question concerns a name for categories with= =20 co-cones over all such diagrams. I don't have a very poetic name for this.= =20 At the moment I'm content with saying that such categories have V-cocones= =20 (or V-cones) depending on directions. I've seen it being called the=20 "Amalgamation Property". But as to what concerns the connection with the question of mixed=20 interchange of limits in Set, one needs to be very careful: If for some doctrine D one defines D-filtered categories to be categories= =20 J such that J-colimits commute with D-limits in Set, then this terminology= =20 will not do, since we then have. 1) If D is equalizers then D-filtered=3Dpseudofiltered. 2) If D is pullback= s=20 then D-filtered=3Dpseudofiltered. 3) If D is pullbacks and terminal objects= ,=20 then D-filtered=3Dfiltered (and not proto-pseudofilterd as one could hope= =20 for) So one need to distinguish between three things: 1) having cocones over certain diagrams. 2) the categories of cocones over certain diagrams are connected.=20 3) commuting in Set with limits over certain diagrams. What has been called "sound doctrines", are the doctrines such that 2) and= =20 3) are equivalent. As a short answer to the open question: If J is a sifted +=20 proto-pseudofiltered category, i.e sifted and span-directed then J is=20 pseudofiltered and connected and thus filtered. (since pseudofiltered=20 categories are categories with filtered connected components) This kind of reflexions and more, with proofs, will soon be available via= =20 my PhD thesis. Best wishes, Marie Bjerrum. 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 > >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 ever= y >> 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/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]