From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6185 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: are fibrations evil? Date: Thu, 16 Sep 2010 22:01:06 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1284773335 26569 80.91.229.12 (18 Sep 2010 01:28:55 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 18 Sep 2010 01:28:55 +0000 (UTC) Cc: categories@mta.ca To: Thomas Streicher Original-X-From: majordomo@mlist.mta.ca Sat Sep 18 03:28:53 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OwmER-0000S2-U8 for gsmc-categories@m.gmane.org; Sat, 18 Sep 2010 03:28:48 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51091) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OwmDI-0007Ko-Jh; Fri, 17 Sep 2010 22:27:36 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OwmDF-0000zY-2E for categories-list@mlist.mta.ca; Fri, 17 Sep 2010 22:27:33 -0300 In-Reply-To: <20100916094755.GA19976@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6185 Archived-At: On Thu, Sep 16, 2010 at 2:47 AM, Thomas Streicher wrote: > For such a thing the key intuitions are lost as > far as I can see (even if I and J are isomorphic the fibre over I may be > inhabited whereas the fibre over J is inhabited). The key necessary change to intuition is that one has to replace "fiber" (itself a non-kosher notion) with "essential fiber". http://ncatlab.org/nlab/show/essential+fiber With this change, all the usual intuitions and facts about fibrations still hold (in corresponding kosher ways). > I doubt that category > theory over a base (topos) can be developed this way. The 2-category of Street fibrations over a given category (such as a topos) is biequivalent to the 2-category of Grothendieck fibrations over that same category, and both are biequivalent to the 2-category of Cat-valued pseudofunctors. (In fact, any Street fibration is equivalent to a Grothendieck fibration, using the same construction which shows that any functor is equivalent to an isofibration; thus the second is a full biequivalent sub-2-category of the first. The second and third are actually strictly 2-equivalent.) Thus, anything kosher that can be done in one can equally be done in the others. > At least it would be > very cumbersome. Has the generalised notion of fibration been used for > something? Indeed it would be cumbersome, and unnecessary for most purposes. The only use I know of for Street fibrations is when working internally to a bicategory. Both Street and Grothendieck fibrations can be defined internally to a strict 2-category, and I believe that if the 2-category has some simple strict 2-limits then every Street fibration will be equivalent to a Grothendieck one, just as in Cat. However, since Grothendieck fibrations are non-kosher, their internal definition involves equality of arrows, and hence is not really sensible in a bicategory rather than a strict 2-category. This was Street's original application. I didn't mean to say that Street's kosher fibrations *should* be used in any place where Grothendieck non-kosher ones suffice, or that the latter aren't easier, simpler, more common, and better to use in practice when possible. This is often the case with kosher and non-kosher things, like weak and strict 2-categories, or bilimits and pseudolimits. But in almost all cases where we use non-kosher things, there *exists* an equivalent kosher notion, and occasionally it happens that the equivalence breaks and in that case we have to use the kosher notion instead. The only thing I was objecting to was your conclusion that equality of objects is sometimes "absolutely necessary" -- in this case, as in many others, it's just very convenient. (There are a few situations where equality of objects -- or, in Toby's language, the use of "strict categories" -- does seem to be conceptually fundamental, such as Peter May's Galois theory example. But I don't think fibrations is one of them.) Why should we distinguish between "absolutely necessary" and "very convenient"? For the "working mathematician" perhaps there is no reason to. But I think that for a category theorist developing new categorical concepts, it is a useful heuristic guide -- if a non-kosher concept is not equivalent to some kosher one, then that is a reason to be suspicious of it, if nothing more. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]