Re: Question on exact sequence

[Steve Lack <s.lack@uws.edu.au>]

Dear Michael,

This is the sort of thing that Enrico Vitale has been working on with
various people for a number of years. I'm sure he'll provide more  precise
references, but the idea is that you think of the vertical morphisms in your
diagrams as internal categories:

A               A+A'
|               | |
| f    <--->    | |
v               v v
A'               A'

(an internal category in Ab amounts to just a morphism - I'll abbreviate
this to just (A,A').)

and then an exact sequence of internal categories, in a suitably defined
sense of exactness, induces a long exact sequence involving the pi_0's and
pi_1's. (pi_0 of an internal category is the cokernel of the corresponding
morphism, while pi_1 is the kernel.)

In your diagram (the "curious" one), the morphism 1:C-->C is saying that
the corresponding internal functor (A,C)-->(B,C) is (not just essentially
surjective but) the identity on objects. This is the relevant notion of
"epi". The morphism 1:A-->A says that the corresponding internal functor
(A,B)-->(A,C) is (among other things) faithful. This is the relevant notion
of "mono". There is also an exactness condition at (A,C).

Vitale, with various coauthors, has studied such exactness conditions at
varying levels of generality, but the simplest of these is just internal
categories in Ab.

Steve.

On 10/11/09 9:57 AM, "Michael Barr" <barr@math.mcgill.ca> wrote:

> I have recently discovered a curious fact about abelian categories.
> First, let me briefly describe the well-known snake lemma.  If we have a
> commutative diagram with exact rows (there are variations without the 0
> at the left end of the top and without the 0 at the right end of the
> bottom, but here is the strongest form)
>
>       0 ---> A ----> B ----> C ----> 0
>              |       |       |
>              |       |       |
>              |f      |g      |h
>              |       |       |
>              v       v       v
>       0 ---> A' ---> B' ---> C' ---> 0
>
> then there is an exact sequence
>   0 --> ker f --> ker g --> ker h --> cok f --> cok g --> cok h --> 0
>
> The curious discovery is that you have any pair of composable maps f: A
> --> B and h: B --> C and you form the diagram (with g = hf)
>                  1       f
>              A ----> A ----> B
>              |       |       |
>              |       |       |
>              |f      |g      |h
>              |       |       |
>              v       v       v
>              B ----> C ----> C
>                  h       1
> you get the same exact sequence.  So I would imagine that there must be
> a "master theorem" of which these are two cases.  Does anyone know what
> it says?  The connecting map here is just the inclusion of ker h into B
> followed by the projection on cok f.
>
> Michael
>
>
>
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