Re: Question on exact sequence

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

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

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "Categories list" <categories@mta.ca>
Sent: Tuesday, November 10, 2009 12:57 AM
Subject: categories: Question on exact sequence


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



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