Date: Tue, 23 May 1995 00:55:53 -0300 (ADT) Subject: [Q] Modified notion of epi Date: Mon, 22 May 1995 12:50:01 -0700 From: Andrew Ensor In any category the pullback of a monic is monic, but this does not hold for epis. However in the category Sets (or in an abelian category) the pullback of a surjective function is surjective. Is there another notion of "epi" for categories that is preserved by pullbacks, and that is surjectivity in the category Sets? Andrew Ensor. Date: Tue, 23 May 1995 09:44:38 -0300 (ADT) Subject: Re: [Q] Modified notion of epi Date: Tue, 23 May 1995 09:38:33 +0000 From: Steve Vickers >Is there another notion of "epi" for categories that is preserved >by pullbacks, and that is surjectivity in the category Sets? Effective descent morphisms. (See Carboni, Janelidze, Kelly and Pare, "On localization and stabilization for factorization systems".) p: A -> B in C (with finite limits) is effective descent iff the pullback functor between slice categories, p*: C/B -> C/A, is monadic. In any topos, effective descent = epi. Steve Vickers. Date: Tue, 23 May 1995 10:42:23 -0300 (ADT) Subject: Re: [Q] Modified notion of epi Date: Tue, 23 May 95 08:40:01 EDT From: Michael Barr The short answer is that only split epis (those with a right inverse) are preserved under pullbacks in "all" categories. In any topos, for example, all epis are stable (the word used when they are preserved by pulling back), but then all epis are regular. Both of these facts are also valid in any abelian category. There is a large class of categories, the so-called regular categories in which regular epis (those that are coequalizers of some pair of arrows) are stable. Small regular categories have a faithful functor into sets^I for some (discrete) index set I that preserves regular epis and finite limits and only regular epis go to epis. (There is also a full, faithful functor into a category of set-valued functors, but that has turned out to be less useful.) One reason that regular categories have turned out to be interesting is that *every* equationally defined category is regular. But, for example, the category of posets and the category of small categories are not regular, whence neither can be defined equationally. Date: Wed, 24 May 1995 00:12:27 -0300 (ADT) Subject: the other short answer Date: Tue, 23 May 1995 10:31:04 -0400 From: Peter Freyd Mike Barr writes: The short answer is that only split epis (those with a right inverse) are preserved under pullbacks in "all" categories. Well, yeah. But there is another short answer. I didn't know how to answer the original note without sounding stupid. I still don't. But there have been those who have given names to just those epis that are preserved under pullback, the most popular being "stable epis." So: The short answer is that only stable epis (those preserved by pullbacks) are preserved under pullbacks. Sorry about this. Anyway: split implies stable. Date: Wed, 24 May 1995 10:53:30 -0300 (ADT) Subject: Re: [Q] Modified notion of epi Date: Wed, 24 May 1995 12:42:30 +1000 From: Ross Street >>Is there another notion of "epi" for categories that is preserved >>by pullbacks, and that is surjectivity in the category Sets? Of course, split epis (= retractions) will answer this if Set has choice. --Ross Date: Mon, 5 Jun 1995 23:35:16 -0300 (ADT) Subject: Re: [Q] Modified notion of epi Date: Mon, 05 Jun 1995 17:40:16 +0200 From: HARDIEKA@uctvms.uct.ac.za I believe Hubertus Bargenda has such a notion. I don't know his current email address but he is at University of the Orange Free State in South Africa.. All best, Keith Hardie.