Re: Question on exact sequence

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

Yes, this is how I understood your original question. In order to make some
comments, let me mention/recall:

You say "a map between two three term sequences" and such a diagram in a
category C is of course the same as a composable pair of morphisms in the
arrow category Arr(C). Therefore let me write it as

f ---> g ---> h   (1)

- and so your question is: when - for an abelian C - does (1) induce an
exact sequence

(0 -->) ker f --> ker g --> ker h --> cok f --> cok g --> cok h (--> 0)
(2)

My comments are:

1. I think Steve made a very good comment. Let me repeat it in my words:
When C is abelian, the category Cat(C) of internal categories in C is
equivalent to Arr(C). Writing cat(f) for the internal category corresponding
to f, we have:

(a) ker f = the internal group of automorphisms of 0 in cat(f) = the
fundamental group of cat(f). Therefore, let me write ker f = pi_1(f).

(b) cok f = the object of connected components of cat(f). Therefore, let me
write cok f = pi_0(f).

After that the sequence (2) becomes

(0 -->) pi_1(f) --> pi_1(g) --> pi_1(h) --> pi_0(f) --> pi_0(g) --> pi_0(h)
(--> 0),

and your question becomes a classical question of abstract homotopy theory!
And Steve gave (a partial?) answer that comes out of Enrico's work.

2. As we both agree, Marco also made a very good comment. Namely, Marco has
a general theorem that applies to (1) in the two cases of main interest: to
the standard snake lemma and to the case you called curious.
.
3. My own comment (although I am slightly changing it here) was/is that the
morphism ker h --> cok f (in (2)) should probably be considered as an extra
structure on (1) rather than something "induced". In Marco's approach it is
an induced morphism on subquotients, but it becomes "induced" in the two
cases of interest for different reasons. And those two reasons ONLY BECOME
the same when we present the two cases of interest as special cases of
Marco's theorem in Marco's way. But using Marco's way we also automatically
equip our diagram with an extra structure. Note also, that an extra
structure appears in Steve's second message as z : A --> C' (which surely
helps to create ker h --> cok f, although I have not checked the details).

4. It is amazing how many non-abelian versions of your question might be
asked! Since Dominique has a non-abelian snake lemma it can be about
semi-abelian/homological/protomodular categories - I would call it
"Bourn-nonabelian" although there are older versions with "old" axioms. Or
it can be Grandis-nonabelian - since Marco has own homological categories,
where he develops abstract homological algebra. Or it can be replacing
(all?) morphisms with pairs of parallel morphisms in a category enriched in
commutative monoids - as I understand this is what Bill had in mind asking
his question. Or it can be replacing f, g, h with internal categories (or
groupoids) in a rather large class of categories - as follows from Steve's
suggestion.

And there are related questions that came out of other interesting
messages...

George

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "George Janelidze" <janelg@telkomsa.net>
Cc: "Categories list" <categories@mta.ca>
Sent: Friday, November 13, 2009 6:06 PM
Subject: Re: categories: Re: Question on exact sequence


> 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
> >

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]