From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/546 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: preprints available Date: Thu, 4 Dec 1997 09:47:48 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017028 26216 80.91.229.2 (29 Apr 2009 14:57:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:57:08 +0000 (UTC) To: categories Original-X-From: cat-dist Thu Dec 4 09:49:17 1997 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id JAA14908; Thu, 4 Dec 1997 09:47:48 -0400 (AST) Original-Lines: 74 Xref: news.gmane.org gmane.science.mathematics.categories:546 Archived-At: Date: Thu, 4 Dec 1997 11:54:52 +0100 From: Marco Grandis The following preprints are now accessible as ps-files, via web of ftp: http://www.dima.unige.it/STAFF/GRANDIS/ ftp://www.dima.unige.it/pub/STAFF/GRANDIS (1). "Limits in double categories", by Marco Grandis and Robert Pare Dbl.Dec97.ps (2). "Weak subobjects and weak limits in categories and homotopy categories", by M.G. Var1.Aug97.ps (3). "Weak subobjects and the epi-monic completion of a category", by M.G. Var2.Dec97.ps *** The first was announced on this mailing list, on 13 Nov 1997. (With respect to the printed preprint, this is a slightly revised version, containing a more detailed comparison with Bastiani-Ehresmann's "limits relative to double categories".) The second and third form an expanded version of a printed preprint ("Variables and weak limits in categories and homotopy categories", Dec 1996), announced on this list on 13 Dec 1996. Abstracts for (2) and (3) are given below. *** (2). Abstract. We introduce the notion of "variation", or "weak subobject", in a category, as an extension of the notion of subobject. The dual notion is called a covariation, or weak quotient. Variations are important in homotopy categories, where they are well linked to weak limits, much in the same way as, in "ordinary" categories, subobjects are linked to limits. Thus, "homotopy variations" for a space S, with respect to the homotopy category HoTop, form a lattice Fib(S) of "types of fibration" over S. Nevertheless, the study of weak subobjects in ordinary categories, like abelian groups or groups, is interesting in itself and relevant to classify variations in homotopy categories of spaces, by means of homology and homotopy functors. (To appear in: Cahiers Top. Geom. Diff. Categ.) (3). Abstract. Formal properties of weak subobjects are considered. The variations in a category X can be identified with the (distinguished) subobjects in the epi-monic completion of X, or Freyd completion FrX, the free category with epi-monic factorisation system over X, which extends the Freyd embedding of the stable homotopy category of spaces in an abelian category (P. Freyd, Stable homotopy, La Jolla 1965). If X has products and weak equalisers, as HoTop and various other homotopy categories, FrX is complete. If X has zero-object, weak kernels and weak cokernels, as the homotopy category of pointed spaces, then FrX is a "homological" category. Finally, if X is triangulated, FrX is abelian and the embedding X --> FrX is the universal homological functor on X, as in the original case. These facts have consequences on the ordered sets of variations. Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.10.353 6805 fax: +39.10.353 6752 http://www.dima.unige.it/STAFF/GRANDIS/