From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4055 Path: news.gmane.org!not-for-mail From: wlawvere@buffalo.edu Newsgroups: gmane.science.mathematics.categories Subject: Re: Comma categories Date: Fri, 02 Nov 2007 12:12:28 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019692 11510 80.91.229.2 (29 Apr 2009 15:41:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:41:32 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Nov 2 16:23:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Nov 2007 16:23:46 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Io23q-0003Kq-6v for categories-list@mta.ca; Fri, 02 Nov 2007 16:20:06 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 7 Original-Lines: 94 Xref: news.gmane.org gmane.science.mathematics.categories:4055 Archived-At: Dear Uwe You are right in thinking that there should be such=20 an exposition because the construction is explicitly=20 or implicitly involved in so many contexts that a=20 formal summary would be useful. Unfortunately,=20 I know of no such exposition though Hugo Volger=20 started one many years ago. As you can see from the TAC Reprint of my=20 thesis, the original motivation was to=20 be able to state the definition of adjointness in a=20 wholly elementary way for arbitrary categories=20 without involving enrichments in some fixed category of sets. If A is a reflective subcategory in=20 some X and if B is coreflective in the same X, then=20 composing the implicit functors yields an adjoint=20 pair between A and B. The point is that conversely=20 any adjoint pair can be so factored through a third=20 "adjunction" category X and the universally available=20 choice has this simple construction as a pullback. It proved to be the appropriate tool for calculating Kan extensions, adequacy comonads, fibrations,etc. Grothendieck defined slice categories and Artin the=20 gluing, both of which are special cases of this construction. Although inserters are interdefinable (like equalizers vs pullbacks), some consider inserters more basic:=20 given x:A->C and y:B->C, one can take the=20 inserter of the two composites AxB->C to obtain=20 the construction under discussion.=20 In the special case A=3DB=3D1 (when the inserter and the=20 "comma" category are the same) we obtain the homset=20 (x,y) of two objects of C. The latter was the reason=20 for my notation: it generalizes a frequent notation for=20 hom.[Recall that every object belongs to a unique=20 category; thus the standard notation C(x,y) is actually redundant (if C is not enriched), though easier to understand. Either notation is preferable to the=20 excessive HomsubC, a back formation not be confused with the informative HomsubR when C arises from=20 adjoining some additional structure R to a given base.] =20 Concerning the bizarre name: (1) I had neglected to give the construction any name,=20 so (2) one started giving it a name based on reading=20 aloud the notation: x comma y; (3) some continued the "name" but changed the notation to a vertical arrow. Since it is well justified to name a category for its=20 objects, and since the effect of insertion is to create=20 objects with one ingredient more of structure, recent=20 discussions here have proposed the name/notation Map(x,y) [or for emphasis Map(subC)(x,y)] for the category with its faithful functor to AxB. Although I often use the word "map" interchangeably with "morphism", note that the above suggests a more concrete content: philosophically, in order to confront=20 objects in two categories A and B, it is necessary to=20 first functorially transport them into a common=20 category C. For example to map a 2-truncated simplicial=20 set to a diffentiable manifold (such as a piece of paper) one first interprets each in appropriate ways as=20 topological spaces, and the resulting objects form a=20 category (having full subcategories of "cartographical"=20 interest). =20 I would be happy to offer a prize for the best exposition! Bill Quoting Uwe Egbert Wolter : > Dear all, >=20 > I'm looking for a comprehensive exposition of definitions and > results > around comma/slice categories. Especially, it would be nice to have > something also for non-specialists in category theory as young > postgraduates. Is there any book or text you would recommend? >=20 > Best regards >=20 > Uwe Wolter >=20 >=20 >=20 >=20