Re: Question on exact sequence

[F William Lawvere <wlawvere@buffalo.edu>]

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 
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 definition
(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 
a resulting map from the right adjoint to the left; the image H of that 
map is a further invariant of objects in the bigger category, recorded in  
the smaller. 

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 
the image exist (in the traditional example, let C be a generic sequence 
and let D be the sequence of zeroes; restricting to the part of A^C where 
d^2=0 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 example
(reflexive graphs) also satisfies the "Nullstellensatz " of my 2007 TAC paper
on cohesion, which is just a way of saying that H reduces to the right adjoint
 itself.

For (nonreflexive) iterated graphs , I.e., for  C a sequence of parallel pairs,
(for example sums of front vs back faces of cubes), an interesting subcategory
of the functor category is the part where the two are equal. This may be useful
for homology of additive objects where rigs of coefficients are not necessarily
rings.

How can exactness and "long exact sequences" be meaningfully  treated for such
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.
> 
> 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.
> 
> 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.
> 
> Michael
> 
> On Wed, 11 Nov 2009, George Janelidze wrote:
> 
> > The "curious discovery" is Exercise 6
> at the end of Chapter VIII ("Abelian> Categories") of Mac Lane's "Categories
> 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 unnaturality of
> d does not make big> problems in concrete situations.
> >
> > ANSWER 2. ker f --> ker g --> ker h   and 
>  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 such a
> canonical d (in the abelian case)> was a big original reason for developing what we
> call today "calculus of> relations" (at the beginning with great
> participation of Saunders himself).>
> > And... in the "curious case = Exercise
> 6" the "canonical d" does not work!> For, consider the simplest case of the 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
> 
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> 
> 
> 
> 



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