Re: Question on exact sequence

[Michael Barr <barr@math.mcgill.ca>]

I do appreciate the example since I wondered if the "connecting
homomorphism" could be induced by a composite of relations as in the snake
lemma.  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 the
replies, but when I have time, I plan to collect them all and try to see
if 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 Marco
mentioned.

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 to
formulate 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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