From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9238 Path: news.gmane.org!.POSTED!not-for-mail From: David Yetter Newsgroups: gmane.science.mathematics.categories Subject: Re: Functors arising from a relational Grothendieck construction Date: Wed, 14 Jun 2017 01:41:04 +0000 Message-ID: References: Reply-To: David Yetter NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1497461035 17892 195.159.176.226 (14 Jun 2017 17:23:55 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Wed, 14 Jun 2017 17:23:55 +0000 (UTC) To: "categories@mta.ca" , Luc Pellissier Original-X-From: majordomo@mlist.mta.ca Wed Jun 14 19:23:51 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1dLC1G-0004Or-SW for gsmc-categories@m.gmane.org; Wed, 14 Jun 2017 19:23:51 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:33565) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dLC20-0003wq-5W; Wed, 14 Jun 2017 14:24:36 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dLC0r-0005rG-TB for categories-list@mlist.mta.ca; Wed, 14 Jun 2017 14:23:25 -0300 In-Reply-To: Accept-Language: en-US Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9238 Archived-At: Dear Luc, Is that all you want, or would you like k and l to be unique, or unique up = to isomorphism in the sense that there is an isomorphism across the diagona= l of the=20 commutative in C created by two such pairs making the whole diagram commute= ? If so, for unique up to isomorphism, such a functor is called a Conduch\'{e= } fibration, and for unique, it is called a discrete Conduch\'{e} fibration= .=A0 There is a discussion of these and related notions in the n-Lab articl= e on Conduch\'{e} functors: https://ncatlab.org/nlab/show/Conduch%C3%A9+functor Best Thoughts, David Yetter Professor of Mathematics Kansas State University P.S. Not in reply to the question. I'd be interested if anyone knows nice = constructions of discrete Conduch\'{e} fibrations. It turns out that a dis= crete Conduch\'{e} fibration over a category with all arrows monic satisfyi= ng the right Ore condition (all cospans complete to commutative squares) wi= th another lifting property, all functors induced on slice categories spli= t, are the ingredients for a construction of C*-algebras generalizing the p= opular graph and k-graph C*-algebras of Raeburn, Kumjian, Pask and their sc= hool.=20 From: Luc Pellissier Sent: Monday, June 12, 2017 4:37 AM To: categories@mta.ca Subject: categories: Functors arising from a relational Grothendieck constr= uction =A0 =20 Dear Category Theorists, with my adviser Damiano Mazza and his other student Pierre Vial, we are loo= king for a name =96 or even better, a reference =96 for the following kind of fu= nctors: Let C and B be two categories, F : C ---> D a functor satisfying, for all morphisms f:c -> c' in C: - if Ff =3D g \circ h, then there exists two morphisms k,l such that =A0 + f =3D k \circ l=20 =A0 + Fk =3D g =A0 + Fl =3D h - if Ff =3D id_a for a certain object a, then f itself is an identity. These functors arise when applying the Grothendieck construction to relatio= nal presheaves: P : B ---> Rel. Indeed, the category of relational presheaves o= n B is equivalent (through the Grothendieck construction) to a category whose objects are such functors over B. If anyone could point us in a right direction, it would be much appreciated= . Best, =97 Luc [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Categories Home Page www.mta.ca Using the list: Articles for posting should be sent to categories@mta.ca Ad= ministrative items (subscriptions, address changes etc.) should be sent to = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]