Re: Question on exact sequence

Re: Question on exact sequence

[Steve Lack]

Dear Michael,

I worked out what exactly Vitale's exactness condition says in this case.
A commutative diagram

   i'    p'
 A'-->B' --> C'
f|   g|     h|
 v    v      v
 A --> B --> C
   i     p

induces a long exact sequence in the way you suggested, when there is
a morphism z:A-->C' for which the induced

0--> A' --> A+B' --> B+C'-->C-->0

is exact.

Steve.


On 10/11/09 2:22 PM, "Steve Lack" <s.lack@uws.edu.au> wrote:

> 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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Re: Question on exact sequence

[George Janelidze]

Dear All,

As I understand, my previous message was written after Marco's and Steve's
messages, although I saw them only afterwards. Unfortunately I don't have
time now, but putting these things together would be very interesting.
Specifically:

1. Marco vs Steve: Marco's result contains what Michael is mentioning
(=Exercise VIII.4.6 in Mac Lane's book) plus Snake Lemma, but does it
contain what Steve describes as Enrico's result?

2. Marco vs George: And do you, Marco, have a canonical connecting morphism?

3. Steve vs George: Do you, Steve - hence Enrico - have a canonical
connecting morphism depending on what you call z : A ---> C' in your (Steve)
message?

George



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

[Marco Grandis]

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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