Re: Question on exact sequence (by G.J.)

[Marco Grandis <grandis@dima.unige.it>]

Dear George,

I receive now your question.

> The Barr's case (=Exercise VIII.4.6 in Mac Lane's book) and the
> Snake Lemma
> seem to have very different canonical connecting morphisms; how
> does your
> (beautiful!) general theorem solve this problem?

All these connecting morphisms are canonically induced on
subquotients, there is
no need of using relations (even though you can, in both cases: a
subquotient
is the same as a subobject in the cat. of relations, and induced
morphisms can
always be computed that way: this is already in Mac Lane's Homology.)

In the Snake Lemma, with Barr's notation:

- Ker h  is a subquotient of  B  (being a subobject of  C),
- Cok f  is a subquotient of  B'  (being a quotient of  A'),
- the connecting morphism is induced by  g: B --> B'.

In the other lemma (Mac Lane, Barr),  Ker h  and  Cok h  are both
subquotients
of the middle object, and the (obvious) connecting morphism is
(trivially) induced
by the identity of the latter.

Subquotients are characterised by a pullback-pushout square
with two monos and two epis (in abelian categories;
more generally in the Puppe-exact ones; more generally in
'my' homological categories, where you do not have relations).
'Regular' induction just means that there is a commutative cube
from the first square to the second.

Best wishes    Marco


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