Re: Question on exact sequence

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

Actually, on further thought, I agree with you.  I didn't originally want
a slick proof but to understand and then I forgot why I really raised the
question.  After all, I had already proved it.  So what I really wanted
and still want to know is what conditions on a map between two three term
sequences gives the 6 term exact sequence (with or without the end 0s).
The situation of the snake lemma is so different from the situation I
(and, obviously others) discovered that one wonders still what general
conditions could possibly encompass the two cases.  That really was my
initial question and that question now comes back to me.

Michael

On Fri, 13 Nov 2009, George Janelidze wrote:

> All right, then I shall better stop, unless there will be new unexpected
> comments (because what Bill and others say, will take us too far...)
>
> George
>
> ----- Original Message -----
> From: "Michael Barr" <barr@math.mcgill.ca>
> To: "George Janelidze" <janelg@telkomsa.net>
> Sent: Friday, November 13, 2009 4:33 AM
> Subject: Re: categories: Re: Question on exact sequence
>
>
>> Actually that diagram with the sums does really answer the question as I
>> had understood it.  There may be a deeper question, but I am not sure how
>> to formulate it.
>>
>> Michael
>>
>> On Fri, 13 Nov 2009, George Janelidze wrote:
>>
>>> 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

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