From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6446 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: source, sinks, and ? Date: Tue, 4 Jan 2011 10:31:19 -0800 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1294191722 26702 80.91.229.12 (5 Jan 2011 01:42:02 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 5 Jan 2011 01:42:02 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Wed Jan 05 02:41:58 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PaINx-0000x2-Db for gsmc-categories@m.gmane.org; Wed, 05 Jan 2011 02:41:57 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:34517) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PaINe-0002Vr-Aa; Tue, 04 Jan 2011 21:41:38 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PaINY-0006TD-L0 for categories-list@mlist.mta.ca; Tue, 04 Jan 2011 21:41:33 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6446 Archived-At: Thanks to everyone who replied. I did intend that the source and target be specified, i.e. to consider, for two given families of objects { x_i }_{i \in I} and { y_j }_{j \in J} (for which either I or J might be empty), a family of morphisms { x_i --> y_j }_{i \in I, j \in J}. This reduces to the notion of sink (resp. cone) described by Reinhard when J (resp. I) is a singleton. "Matrix" and "array" are both good words, although I agree that the non-composability in general makes "matrix" slightly misleading. One might also observe that such a family can be identified with a diagram indexed on the collage (or cograph) of a profunctor/distributor between discrete categories (specifically, the profunctor constant at 1). But that doesn't immediately suggest a conciser name to my mind. One place such families occur is in what one might call "joint source/sink factorization systems". For instance, in Ross Street's paper "The family approach to total cocompleteness and toposes," a "familially regular category" is defined to be one in which any such "array" with J finite factors into a strong-epic sink followed by a monic source, and strong-epic sinks are stable under pullback. Another is that just as the limit of a diagram is a cone over that diagram with a universal property, a *multilimit* of a diagram can be described as an "array" over that diagram (which we may regard as a family of cones with the same codomain) with a universal property. Mike On Tue, Jan 4, 2011 at 2:44 AM, Reinhard Boerger wrote: > Hello, > > I am used to a slightly different terminology, which seems appropriate. > >> A family of morphisms { x_i --> y }_{i \in I} in some category, all >> with the same codomain, is called a "sink" or a "cocone". > > For a sink, as I know it, the codomain should also be specified, i.e. a s= ink > is given by an object y and a family of morphisms x_i --> y. If I is not > empty, this does not matter, but for empty I at least y should be given. = A > cocone is given by an object y and a natural transformation from some > functor to the constant functor with value y; her y is also specified. So= a > sink is essentially a discrete cocone. > > =A0 A family { >> x --> y_j }_{j \in J} all with the same domain is called a "source" or >> a "cone". > > These are the dual notions. > >> Is there a name for a family of the form { x_i --> y_j }_{i >> \in I, j \in J} ? =A0A cylinder? =A0Or a frustrum (since I \neq J)? > > I do not know. Where does it occur? Probably the domain and codomain shou= ld > also be specified, possibly even an arrow. If a non-empty collection of > arrows behave similarly (e.g. is mapped to the same arrow by a given func= tor > F), this means the same a saying that they all behave in the same way as = a > given arrow 8e.g are mapped to some special morphism by F). A collection = of > two objects x,y (prescribed domain and codomain) is something different; = it > does not give an arrow Fx -->Fy. > > > Greetings > Reinhard > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]