From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5729 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: Four problems corrected Date: Sun, 2 May 2010 09:58:18 -0400 Message-ID: References: <20100430211359.nbm6pfhjk0wgkgwc@mail.math.yorku.ca> Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1272888682 28068 80.91.229.12 (3 May 2010 12:11:22 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 3 May 2010 12:11:22 +0000 (UTC) To: , Original-X-From: categories@mta.ca Mon May 03 14:11:19 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1O8uUV-0000eH-D2 for gsmc-categories@m.gmane.org; Mon, 03 May 2010 14:11:15 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1O8tvP-0000gG-4b for categories-list@mta.ca; Mon, 03 May 2010 08:34:59 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5729 Archived-At: Dear All,=20 there was a few typographic errors my last message.=20 I am sending you the correction. Best,=20 Andr=E9 ----------------------------------------------------- =20 Dear Walter, Let me sketch a possible solution to problem 4=20 (along the lines you have suggested). > If P: C --> S is a functor, I denote by V(P) the subcategory of C=20 >which has the same objects and as maps the vertical maps i.e. the f's = such >that P(f) is an identity. Let V be a subcategory of a (small) category = C. > What conditions >must the pair (C,V) satisfy in order that > there exists a fibration P with domain C such that V =3D V(P)? I first make a few observations on the properties of a Grothendieck fibration P:C--->B. 1) If V is the subcategory of vertical maps in C, then a morphism in C is cartesian with respect to the functor P iff it is right orthogonal=20 (=3D it has the unique right lifting property) to every map in V. =20 2) if R(V) is the class of maps in C which are right orthogonal to every map in V, then every map f in C admits a factorisation f=3Dcv with v in V and c in R(V). 3) the base change of a map in V along a map in R(V) exists and can be taken in V (up to an isomorphism).=20 4) A Grothendieck fibration P:C--->B is *connected* if its fibers=20 are connected categories. It is easy to see that every Grothendieck=20 fibration P:C--->B admits a factorisation P=3DSQ:C-->E-->B with Q:C-->E = a connected Grothendieck fibration and S:E-->B a discrete fibration=20 (the fibers of S are the connected components of the fibers of P). The vertical maps of P coincide with the vertical maps of Q. This shows that if the problem has a solution, then there is one in which the fibration P:C-->B is connected, in which case there is a=20 natural bijection between the objects of the=20 base category B and the set connected components of the sub-category of vertical maps. We shall suppose the the conditions 2-3 above are satisfied but we will we will need an extra condition later.=20 The idea then is to declare that the objects of B *are* the connected components of the subcategory V. If C_0 and C_1 are two connected components of V, consider the distributor D:C_0-->C_1 obtained by putting D(a,b)=3DC(a,b) for every object a of C_0 and every object b of C_1.=20 The idea is to put B(C_0,C_1)=3D colimit D and to use the composition of arrows in C for defining the composition of morphisms in B. But we have to make sure that the composition so defined is unambigous. And this is where the extra condition 4 is popping out.=20 First, the distributor D:C_0-->C_1 is locally=20 corepresentable by a family. More precisely, for any object b of C_1 the presheaf=20 D(-,b):C_0-->Set is a coproduct of representable presheaves. This follows directly from condition 2 (the representing objects are morphisms f:a-->b in R(V)). Let us put=20 T(b)=3D\pi_0D(-,b)=3Dcolimit D(-,b) =20 This defines a functor T:C_1--->Set (which depends on C_0). It follows from condition 3 that the functor T inverts every morphism of C_1. This shows that the distribuor D:C_0-->C_1 is of a very special type. The category of elements of the functor T is a *covering* el(T)-->C_1 (a covering is a discrete fibration which is also an opfibration). It follows that the distributor D:C_0-->C_1 can be represented as a span C_0<---el(T)--->C_1 in which the second leg is a covering. Assuming that the conditions 2 and 3 are=20 satisfied, the problem of Benabou will have a solution=20 iff this covering is trivial (ie it is a product) for any pair of connected components C_0 and C_1 of V. This is true for example when the connected components of V are=20 simply connected. Best, Andr=E9 -------- Message d'origine-------- De: tholen@mathstat.yorku.ca [mailto:tholen@mathstat.yorku.ca] Date: ven. 30/04/2010 21:13 =C0: Joyal, Andr=E9 Cc: categories@mta.ca; tholen@mathstat.yorku.ca Objet : Re: categories: Four problems =20 In the article J. Rosicky, W. Tholen, "Factorization, fibration and torsion", J.=20 Homotopy Theory and Related Structures (electronic) 2 (2007) 295-314 we prove a result closely related to Problems 3 and 4 below (variations=20 of which may well have appeared earlier?), as follows: In a finitely complete category C, (E,M) is a simple reflective=20 factorization system of C (in the sense of Cassidy, Hebert, Kelly, J.=20 Austr. Math. Soc 38, (1985)) if, and only if, there exists a=20 prefibration P: C ---> B preserving the terminal object of C with E =3D=20 P^{-1}(Iso B) and M =3D {P-cartesian morphisms}. Here "prefibration" means that for all objects c in C, the functors C/c=20 ---> B/Pc induced by P have right adjoints, such that the induced=20 monads are idempotent. (For a fibration one asks the counits to be=20 identity morphisms.) Of course, Jean's question wants P^(-1)(Iso) to be=20 replaced by the non-iso-closed class P^(-1)(Identities), which prevents=20 the class from being part of an ordinary factorization system. But=20 (without having looked into this at all) I would suspect that there is=20 probably a (more cumbersome) reformulation of the theorem above which=20 would address that concern. Regards, Walter. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]