Date: Mon, 23 Sep 1996 13:59:07 -0300 (ADT) Subject: irreducibility Date: Mon, 23 Sep 1996 17:31:19 +0200 From: Pierre Ageron Say that an object X in a category is irreducible iff Hom(X,-) preserves pushouts. Probably this (or a similar) notion is classical. Am I right ? PIERRE AGERON 1) coordonnees bureau adresse : mathematiques, Universite de Caen, 14032 Caen Cedex telephone : 02 31 56 57 37 telecopie : 02 31 93 02 53 adresse electronique : ageron@math.unicaen.fr 2) coordonnees domicile adresse : 28 rue de Formigny 14000 Caen telephone : 02 31 84 39 67 Date: Mon, 23 Sep 1996 15:22:33 -0300 (ADT) Subject: Re: irreducibility Date: Mon, 23 Sep 1996 13:55:53 -0400 (EDT) From: F William Lawvere Concerning Pierre Ageron's proposed definition of "irreducible" : Since such terms as "connected"," local", etc should always exclude the empty case, "connected" in particular is usually defined to mean preserving all coproducts (including the empty one) . Preserving moreover pushouts would then amount to "connected and projective", which in a presheaf category catches only the Cauchy completion of the representables. Usually "irreducible" is understood as something broader than that; for example among finite presheaves on a finite category, there are an infinite number of connected objects iff the finite category is not a groupoid. Indeed in G-sets for G a group, each orbit is connected, but the only orbit which is projective is the biggest one. Date: Thu, 26 Sep 1996 12:09:24 -0300 (ADT) Subject: Re: irreducibility Date: Thu, 26 Sep 1996 11:33:11 +0200 From: Pierre Ageron As William Lawvere pointed out courteously, my tentative definition of "irreducible" was stupid. So let me ask: is there a standard categorical treatment of irreducibility (or, dually, of primality), or is it an essentially order-theoretic concept? Pierre Ageron PIERRE AGERON 1) coordonnees bureau adresse : mathematiques, Universite de Caen, 14032 Caen Cedex telephone : 02 31 56 57 37 telecopie : 02 31 93 02 53 adresse electronique : ageron@math.unicaen.fr 2) coordonnees domicile adresse : 28 rue de Formigny 14000 Caen telephone : 02 31 84 39 67