Re: Question on exact sequence

["George Janelidze" <janelg@telkomsa.net>]

I have further comments to Marco and Steve (maybe tomorrow...), but now I am
only answering

> 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.

from Michael's message:

Dear Michael,

Does "formulate" mean "obtain/deduce"? Obtaining the curious sequence as a
consequence of the snake lemma is actually easy, and Saunders surely knew
it - which probably explains why did not he raise your question. Given your
f : A ---> B, h : B ---> C and g = hf, just apply the snake lemma to

    <1,f>         [f,-1]
A ---> A + B ---> B
 |              |              |
 | f            | g+1       | h
v             v             v
B ---> C + B ---> C
    <h,1>        [1,-h]

where + denotes the direct sum, <...> "uses" it as product, and [...] "uses"
it as coproduct (and use the fact that Ker(g) = Ker(g+1)).

However, this does not answer your original question of course.

George

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "George Janelidze" <janelg@telkomsa.net>; <categories@mta.ca>
Sent: Thursday, November 12, 2009 2:41 PM
Subject: categories: Re: Question on exact sequence


> 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
>

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