From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6447 Path: news.gmane.org!not-for-mail From: JeanBenabou Newsgroups: gmane.science.mathematics.categories Subject: Re: source, sinks, and ? Date: Wed, 5 Jan 2011 05:30:51 +0100 Message-ID: References: Reply-To: JeanBenabou NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1294280054 10398 80.91.229.12 (6 Jan 2011 02:14:14 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 6 Jan 2011 02:14:14 +0000 (UTC) To: Michael Shulman , Categories Original-X-From: majordomo@mlist.mta.ca Thu Jan 06 03:14:09 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 1PafMf-0007vJ-0H for gsmc-categories@m.gmane.org; Thu, 06 Jan 2011 03:14:09 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42916) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PafMF-0007sU-9X; Wed, 05 Jan 2011 22:13:43 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PafLp-0007Po-WF for categories-list@mlist.mta.ca; Wed, 05 Jan 2011 22:13:18 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6447 Archived-At: In his mail, Mike Shulman wrote, > 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. 1 - In the spirit of the word "array", which I proposed, I suggest the following names for two special cases. (i) When I = 1 , instead of "cone", "column" (ii) When J = 1 , instead of "sink", "row" This would have the following advantages: (a) In the case of "matrices", i.e. when the product is defined, it would fit with the usual matrix terminology. (b) We wouldn't have to change our use of "cone" and "co-cone" over a diagram D, rows and columns would be the special cases, when D is discrete. I have often used "rows" and "columns" in the context of general "matrices", which I explained in my previous mail, without having met any ambiguity or incompatibility > 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. 2- This "ad hoc" identification, apart from the fact that it "doesn't immediately suggest a conciser name", needs complicated notions such as distributors and collages. Moreover it "doesn't immediately suggest" generalizations. There is a very simple interpretation in terms of the canonical fibration Fam(C) --> Set which can be easily generalized, and permits to define "arrays" for arbitrary fibrations p: X --> S, provided S has finite products. With mild assumptions on p and X, one can even define "matrices" and develop a "matrix calculus" Best to all, Jean > 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]