From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5258 Path: news.gmane.org!not-for-mail From: F William Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Thu, 12 Nov 2009 21:05:51 -0500 Message-ID: <54845.1258077951@buffalo.edu> Reply-To: F William Lawvere NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1258079770 20765 80.91.229.12 (13 Nov 2009 02:36:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 13 Nov 2009 02:36:10 +0000 (UTC) To: "George Janelidze" , , "Michael Barr" Original-X-From: categories@mta.ca Fri Nov 13 03:36:03 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1N8m14-0006Cg-2I for gsmc-categories@m.gmane.org; Fri, 13 Nov 2009 03:36:02 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8lcI-0006uB-Uo for categories-list@mta.ca; Thu, 12 Nov 2009 22:10:27 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5258 Archived-At: The remarks of Clemens together with Murray Adelman's construction as alluded to by Ross suggest the following. The 1960 triumph of abelian categories was followed by a decade which Barr and Grothendieck showed that exactness=20 has little to to with additivity. But homology itself would seem also to have little to do with additivity, if we take seriously the following defin= ition (that is actually mentioned in passing in many books without the specific mention of that other 50-year old triumph of category theory). Given a full inclusion that has both left and right adjoints, there is=20 a resulting map from the right adjoint to the left; the image H of that=20 map is a further invariant of objects in the bigger category, recorded in = =20 the smaller.=20 For example A^C will have a full subcategory determined by a given surjective functor C->D so if A is complete the two adjoints and=20 the image exist (in the traditional example, let C be a generic sequence=20 and let D be the sequence of zeroes; restricting to the part of A^C where= =20 d^2=3D0 may make computing H easier but will not change the definition). Of course if the left adjoint preserves products, then so will H and hence H will preserve any kind of algebraic structure. But the simplest ex= ample (reflexive graphs) also satisfies the "Nullstellensatz " of my 2007 TAC pap= er on cohesion, which is just a way of saying that H reduces to the right adjo= int itself. For (nonreflexive) iterated graphs , I.e., for C a sequence of parallel pa= irs, (for example sums of front vs back faces of cubes), an interesting subcateg= ory of the functor category is the part where the two are equal. This may be us= eful for homology of additive objects where rigs of coefficients are not necessa= rily rings. How can exactness and "long exact sequences" be meaningfully treated for s= uch functors H in non-abelian contexts ? Bill On Thu 11/12/09 7:41 AM , Michael Barr barr@math.mcgill.ca sent: > I do appreciate the example since I wondered if the "connecting > homomorphism" could be induced by a composite of relations as in the > snakelemma. I thought not and George has provided an example. Since > Tuesday,we have had house guests so I really have not had time to absorb = all > thereplies, but when I have time, I plan to collect them all and try to > seeif there is a satisfactory general answer of which the two instances I > described are special cases. There is something going on here that I > don't quite comprehend (although maybe the answer is in the theorem > Marcomentioned. >=20 > Since my curious sequence was an exercise in CWM, it is surprising that > Saunders never raised the question in the form I did. The conclusion > certainly looks like something out of the snake lemma, but I was unable > toformulate it as a cosequence. >=20 > Incidentally, the theorem on acyclic models, as it appears in my book, > can be described as a map induced by a composite of relations that, in > homology, becomes functional. >=20 > Michael >=20 > On Wed, 11 Nov 2009, George Janelidze wrote: >=20 > > The "curious discovery" is Exercise 6 > at the end of Chapter VIII ("Abelian> Categories") of Mac Lane's "Categor= ies > for the Working Mathematician"...> > > However, I think it is an interesting question, > and:> > > When for the standard snake lemma Michael says > "...there is an exact> sequence > > 0 --> ker f --> ker g --> ker h --> > cok f --> cok g --> cok h --> 0", what> does "there is" mean? > > > > There are two well known answers: > > > > ANSWER 1. ker f --> ker g --> ker h and > cok f --> cok g --> cok h are> the obvious induced morphisms and there = exists a > "connecting morphism" d :> ker h ---> cok f making the sequence above > exact. Such a d is not unique:> for instance if d is such, then so is -d. > However, since the snake lemma> holds in functor categories, the unnatura= lity of > d does not make big> problems in concrete situations. > > > > ANSWER 2. ker f --> ker g --> ker h and=20 > cok f --> cok g --> cok h are> the obvious induced morphisms as before, = while > THE "connecting morphism" d :> ker h ---> ker f is the composite of the > zigzag> > > ker h ---> C <--- B ---> B' <---A' > ---> cok f> > > (where the arrows are considered as internal > relations). This "canonical> connecting morphism" d works even in the > non-abelian case of Dominique Bourn> as I learned from my daughter Tamar = who > developed the "relative version".> Note also, that the desire to have suc= h a > canonical d (in the abelian case)> was a big original reason for developi= ng what we > call today "calculus of> relations" (at the beginning with great > participation of Saunders himself).> > > And... in the "curious case =3D Exercise > 6" the "canonical d" does not work!> For, consider the simplest case of t= he composite > 0 ---> B ---> 0: the exact> ker-cok sequence will become > > > > 0 --> 0 --> 0 --> B --> B --> 0 > --> 0 --> 0,> > > where B --> B must be an isomorphism, while > it is easy to check that the> "canonical d" will become the relation > opposite to the zero morphism B -->> B. > > > > A possible conclusion is that the "master > theorem" should involve some kind> of "d" as an extra > structure.> > > To Steve's message: does Enrico really > generalize the standard snake lemma> and the "curious case" > simultaneously?> > > George >=20 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] >=20 >=20 >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]