From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8284 Path: news.gmane.org!not-for-mail From: Zhen Lin Low Newsgroups: gmane.science.mathematics.categories Subject: A condition for functors to reflect orthogonality Date: Mon, 4 Aug 2014 19:24:02 +0100 Message-ID: Reply-To: Zhen Lin Low NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1407249472 2756 80.91.229.3 (5 Aug 2014 14:37:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 5 Aug 2014 14:37:52 +0000 (UTC) To: categories list Original-X-From: majordomo@mlist.mta.ca Tue Aug 05 16:37:41 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 1XEfru-0003so-Dk for gsmc-categories@m.gmane.org; Tue, 05 Aug 2014 16:37:38 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50335) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XEfrM-0003Rm-KQ; Tue, 05 Aug 2014 11:37:04 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XEfrL-0006No-GW for categories-list@mlist.mta.ca; Tue, 05 Aug 2014 11:37:03 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8284 Archived-At: Dear categorists, I am wondering if the following property of a functor U : C -> D has a name in the literature: * 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. More precisely: * For any morphisms X -> Y and Z -> W in C, the induced commutative diagram 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) is a pullback square. 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. 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". Best wishes, -- Zhen Lin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]