From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3069 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Undirected graph citation Date: Sun, 5 Mar 2006 21:15:01 +0200 Message-ID: <005d01c64089$1b437220$0b00000a@C3> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019077 7125 80.91.229.2 (29 Apr 2009 15:31:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:31:17 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Mon Mar 6 09:27:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Mar 2006 09:27:33 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FGFgU-00030s-Up for categories-list@mta.ca; Mon, 06 Mar 2006 09:23:34 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 15 Original-Lines: 221 Xref: news.gmane.org gmane.science.mathematics.categories:3069 Archived-At: Dear Bill, Indeed, there were no monoids in Vaughan's original message of February 28, but since you have mentioned them in your message of March 1, and since you were talking there about "...lacuna of explicitness ... in many papers on Galois theory...", I simply wanted to say that: I do not see any relevance of these kinds of presentations in Galois theory (apart from the fact the internal pre-whatever-s in a category X form an X-valued presheaf category). On the other hand Galois theory is not the end of the World, and I think the beauty and importance of those your ideas is clear to everyone who saw them. Putting myself in risk of making my message boring, I would like to make one more remark concerning your last message and Galois theory: You say: "...applying a non-exact functor F to a group..." - true and fine, but I have actually mentioned F(R) for R being not a group, but another extreme case of a groupoid, namely an equivalence relation. What seems to be most amazing is, that, because F preserves not-all-but-some pullbacks, there are beautiful examples where R is an equivalence relation and F(R) is a group; in simple words, F creates a group out of nothing! The classical example, as you know, is: if R is the kernel pair of a universal covering map E ---> B of a "good" connected topological space B, and F is the functor sending ("good") topological spaces to the sets of their connected components, then F(R) is the fundamental group of B. The same thing is true in other Galois theories of course. George ----- Original Message ----- From: "F W Lawvere" To: Sent: Sunday, March 05, 2006 3:21 AM Subject: categories: Re: Undirected graph citation > > Dear George, > > Concerning undirected graphs, the Boolean algebra classifier, and > the intermediate sub-topos that suffices for groupoids: > > The special feature of these toposes I wanted to emphasize is not that > some of them can be generated by monoids, but rather (whether one splits > idempotents or not) that the site of operators is itself a full > subcategory of the category of sets. This is a small part of the point > that Vaughn wanted to make, I believe. Having the direct visualization of > this system of operators available as merely maps between certain small > sets is a useful auxiliary to formal presentations of the d,s kind. > As you mention, a basic way in which such presheaves can arise is by > applying a non-exact functor F to a group; the fact that the exponents on > the group are just these ordinary sets explains why we obtain an object > in this sort of topos (which can serve as a presentation of another group, > if desired). > > As noted in my unpublished (but widely distributed) paper on toposes > generated by codiscrete objects, the Yoneda embedding in these cases > produces of course a full sub-category of a topos, one which looks > exactly like (a piece of) the category of sets; of course this is not > the discrete inclusion, but its dialectical opposite, the codiscrete one. > > Bill > ************************************************************ > F. William Lawvere > Mathematics Department, State University of New York > 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA > Tel. 716-645-6284 > HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere > ************************************************************ > > > > On Fri, 3 Mar 2006, George Janelidze wrote: > > > I am not sure if I really understand what is the target of this discussion, > > but I would like to make some comments to Bill's messages: > > > > The Poincare' groupoid is (up to an equivalence) nothing but the largest > > Galois groupoid, and it is directly available as soon as one has what I call > > Galois structure in my several papers, if we assume that every object of the > > ground category has a universal covering. This is certainly the case for > > every locally connected topos with coproducts and enough projectives. > > Therefore this is certainly the case for every presheaf topos. Therefore > > what Bill means by "directly available" should be not "available without > > going through geometric realization" but just "can be calculated as the > > result of reflection" (probably this is exactly what Bill had in mind). > > > > Moreover, it was Grothendieck's observation that Galois/fundamental > > groupoids are to be defined as quotients of certain equivalence relations - > > in fact kernel pairs, and this observation was used by many authors in topos > > theory and elsewhere; my own observation (1984) then was that one can make > > Galois theory purely categorical by using not "quotients" but "images under > > a left adjoint" (the first prototype for me was actually not Grothendieck's > > but Andy Magid's "componentially locally strongly separable" Galois theory > > of commutative rings). What I am trying to conclude is that the > > Galois/fundamental groupoids actually arise not from anything simplicial but > > from abstract category theory: it is just a result of a game with adjoint > > functors between categories with pullbacks. > > > > In another message Bill says: "A similar lacuna of explicitness occurs in > > many papers on Galois theory where pregroupoids are an intermediate step ; > > the description of the pregroupoid concept is really just a presentation > > of the monoid of endomaps of the 4-element set..." Assuming that everyone > > understands that this is not about classical Galois theory (I don't think > > somebody like J.-P. Serre ever mentions pregroupoids) and not about what > > Anders Kock calls pregroupoids, let me again return to the categorical > > Galois theory: > > > > If p : E ---> B is an "extension" in a category C, R its kernel pair, and F > > : C ---> X the left adjoint involved in a given Galois theory, then one > > wants to define the Galois groupoid Gal(E,p) as F(R) = the image of R under > > F (I usually write I instead of F, but in an email message this does not > > look good...). But if our extension p : E ---> B is not normal, then, since > > F usually does not preserve pullbacks, F(R) is not a groupoid - it is a > > weaker structure, the "equational part" of groupoid structure. This weaker > > structure is still good enough to define its internal actions in X and these > > internal actions classify covering objects over B split by (E,p). Hence this > > weaker structure needs a name and I called it "pregroupoid" (I did not know > > that this term was already overused for almost the same and for unrelated > > concepts). I cannot speak for everyone, but for my own purposes there are > > actually several possible candidates for the notion of pregroupoid and half > > of them can certainly be defined as monoid actions for a specific monoid, > > like the one Bill mentions. However, in each case we deal with a "very > > small" category actions and it is a triviality to observe that that category > > can be replaced with a monoid. Essentially, what you need is to check that > > the terminal object (in your category of pregroupoids) has either no proper > > subobjects or only one such, which must be initial. In this observation - > > due to Max Kelly, about the categories monadic over powers of Sets being > > monadic over Sets, one usually says "strictly initial"; but we can omit > > "strictly" here since it is about a topos. > > > > George Janelidze > > > > ----- Original Message ----- > > From: "F W Lawvere" > > To: > > Sent: Thursday, March 02, 2006 8:32 PM > > Subject: categories: Re: Undirected graph citation > > > > > > > > > > As Clemens Berger reminds us, the category of small categories > > > is a reflective subcategory of simplicial sets, with a reflector that > > > preserves finite products. But as I mentioned, there is a similar > > > "advantage" for the Boolean algebra classifier (=presheaves on non-empty > > > finite cardinals, or "symmetric" simplicial sets): > > > The category of small groupoids is reflective in this topos, with the > > > reflector preserving finite products. Thus the Poincare' groupoid of a > > > simplicial complex is directly available. (The simplicial complexes are > > > merely the objects generated weakly by their points, a relation which > > > defines a cartesian closed reflective subcategory of any topos.) > > > > > > It is not clear how one is to measure the loss or gain of combinatorial > > > information in composing the various singular and realization functors > > > between these different models. Is there such a measure? > > > > > > > > > Bill Lawvere > > > > > > ************************************************************ > > > F. William Lawvere > > > Mathematics Department, State University of New York > > > 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA > > > Tel. 716-645-6284 > > > HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere > > > ************************************************************ > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >