From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/654 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: CATS Are primes ever generators? Date: Mon, 16 Feb 1998 10:26:09 +1100 (EST) Message-ID: <199802152326.KAA16703@milan.maths.su.oz.au> References: <199802131712.JAA10069@blackhawk.kestrel.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241017101 26742 80.91.229.2 (29 Apr 2009 14:58:21 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:58:21 +0000 (UTC) Cc: categories@mta.ca, espinosa@kestrel.edu To: espinosa@kestrel.edu Original-X-From: cat-dist Mon Feb 16 13:40:10 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA21238 for categories-list; Mon, 16 Feb 1998 10:06:28 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-reply-to: <199802131712.JAA10069@blackhawk.kestrel.edu> (message from David Espinosa on Fri, 13 Feb 1998 09:12:18 -0800) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 56 Xref: news.gmane.org gmane.science.mathematics.categories:654 Archived-At: > X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f > Date: Fri, 13 Feb 1998 09:12:18 -0800 > From: David Espinosa > Cc: espinosa@kestrel.edu > Precedence: bulk > > > > We can say that an object P in a category with coproducts is *prime* > if whenever f : P -> A+B, f factors through one of the injections into > A+B. > > (1) I didn't find any reference to this (obvious) notion of > primality in the standard texts. Does it occur anywhere? > > (2) Is there any condition on the category under which the set of > primes is a generating family? Since objects are decomposable > into a "quotient of a coproduct of generators" (Borceux, volume 1, > page 151), this would give a decomposition into primes. > > Thanks, > > David > > > Dear David, One convenient setting for your question is provided by _extensive_ categories (see the paper ``Introduction to extensive and distributive categories'' by Carboni, Lack, and Walters, appearing in JPAA 1993). A category E with finite coproducts is said to be extensive if for all objects x,y of E, the ``coproduct functor'' E/x x E/y --> E/(x+y) is an equivalence. For such a category E, an object p is prime in your sense if and only if it is connected, i.e. if and only if it admits no proper coproduct decomposition; this in turn is equivalent to the representable functor E(p,-):E-->Set preserving coproducts. An example of an extensive category is given by Fam(C) for C a (small) category. The objects of Fam(C) are finite families (C_i)_{i\in I} and an arrow from (C_i)_I to (D_j)_J comprises a function f:I-->J and a family of arrows C_i-->D_fi in C. Fam(C) is the free category with finite coproducts on the category C. The connected (=prime) objects are precisely the singleton families. One can characterize the categories of the form Fam(C) as those extensive categories with a small set of connected objects such that every object is a finite coproduct of connected objects. (It seems possible that you could replace ``extensive'' in the last sentence by ``category with finite coproducts'' provided that you also replace ``connected'' by ``prime and connected'', but I haven't thought about this.) Best wishes, Steve.