From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8287 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Re: A condition for functors to reflect orthogonality Date: Wed, 6 Aug 2014 05:08:56 +0200 Message-ID: References: Reply-To: =?iso-8859-1?Q?Jean_B=E9nabou?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1283) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1407372750 28347 80.91.229.3 (7 Aug 2014 00:52:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 7 Aug 2014 00:52:30 +0000 (UTC) Cc: categories list To: Zhen Lin Low , Original-X-From: majordomo@mlist.mta.ca Thu Aug 07 02:52:25 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XFBwN-0008AB-Oz for gsmc-categories@m.gmane.org; Thu, 07 Aug 2014 02:52:23 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50632) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XFBvs-0000xE-T4; Wed, 06 Aug 2014 21:51:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XFBvt-0006Ue-FN for categories-list@mlist.mta.ca; Wed, 06 Aug 2014 21:51:53 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8287 Archived-At: Dear Zhen Lin, =20 Before I answer your question let me give names to arbitrary maps of C: f: X --> Y, g: Z --> W, x: Z -->X, y: W --> Y and h: W --> X . Your = condition reads: for every pair (f,g) and every triple (x,y,h') where: yg =3D fx and h': UW --> UX satisfies : h' Ug =3D Ux and Uf h' =3D = Uy, there exits a unique h: W -->X such that: Uh =3D h', hg =3D x and fh = =3D y .=20 One verifies first that this condition is satisfied for all pairs (f,g) = iff it is satisfied for f arbitrary and g is an identity. In that case that means that f is what I call hypercartesian (which in = the anglo-american literature is called cartesian). Since f is arbitrary = , the condition becomes:=20 (i) Every map of C is hyper cartesian. Let me call U locally full and faithful (lff) iff for every object X of = C the obvious functor =20 C/X --> D/UX is full and fathull. In my mail to Joyal and the catgory list, dated July 28 I already = mentioned that (i) is equivalent to (ii) U is lff. I also said, it is obvious, that U full and faithful =3D> U is lff .=20 I mentioned also the case of groupoids, with a sharper result than the = one you stated, namely: If C is a groupoid, every functor U: C --> D, = where D is arbitrary, is lff. Let me add a remark which was not in my mail to Joyal, namely, the = previous property characterizes groupoids. More precisely we have: PROPOSITION 1. Let C be a category. The following are equivalent: (i) C is a groupoid (ii) Every functor with domain C is lff (iii) The unique functor C --> 1 is lff. There are MANY MORE properties of lff functors which would be too long = to give here. Let me mention a few which are not in my mail to Joyal. The following theorem generalizes greatly the previous proposition. THEOREM. Let U: C --> D be a fibration. The following are equivalent: (i) U is lff (ii) All the fibers of U are groupoids. (iii) U is conservative (i.e. reflects isomorphisms). Such fibrations are very important. Because of (ii) they have sometimes = been called groupoid fibrations. In particular, it follows from (iii), = that for such a fibration if D is a groupoid so is C. You said that the functors satisfying your condition are stable by = composition. This result can be strengthened since we have: PROPOSITION 2. Let U and V be functors such that the composite UV is = defined.=20 If U is lff, then UV is lff iff V is. The following result is easy to prove but nevertheless important for = many theoretical reasons. THEOREM 2. lff functors are stable by pull back along any functor. I could add many significant results, in particular about cartesian = functors, or orthogonality but this mail is already a bit long, and I = apologize for this length. Thus there is no need to give a name to the property you mentioned, = locally full and faithful describes precisely this property. Best wishes, Jean=20 Le 4 ao=FBt 2014 =E0 20:24, Zhen Lin Low a =E9crit : > Dear categorists, >=20 > I am wondering if the following property of a functor U : C -> D has a = name > in the literature: >=20 > * For every lifting problem in C and any solution in D to the image = under > U, there is a unique solution in C whose image under U is that = solution. >=20 > More precisely: >=20 > * For any morphisms X -> Y and Z -> W in C, the induced commutative = diagram >=20 > C(W, X) ------> C(Z, X) \times_{C(Z, Y)} C(W, Y) > | | > | | > v v > D(UW, UX) --> D(UZ, UX) \times_{D(UZ, UY)} D(UW, UY) >=20 > is a pullback square. >=20 > Of course, any fully faithful functor has the property in question; a = less > trivial example is the projection from a (co)slice category to its = base. > Every functor between groupoids has this property, so they need not be > faithful. One also notes that the class of functors with this property = is > closed under composition. >=20 > It is not hard to see that if a functor has the above property, then = it > reflects both orthogonality and weak orthogonality in the naive sense. = The > converse is false. Nonetheless, my inclination is to call these = functors > "orthogonality-reflecting". >=20 > Best wishes, > -- > Zhen Lin >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]