From: Steve Lack <s.lack@uws.edu.au>
To: Michael Barr <barr@math.mcgill.ca>, <categories@mta.ca>
Subject: Re: Question on exact sequence
Date: Wed, 11 Nov 2009 22:05:46 +1100 [thread overview]
Message-ID: <E1N8Ehb-0005cT-2y@mailserv.mta.ca> (raw)
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
>>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next reply other threads:[~2009-11-11 11:05 UTC|newest]
Thread overview: 21+ messages / expand[flat|nested] mbox.gz Atom feed top
2009-11-11 11:05 Steve Lack [this message]
2009-11-11 16:36 ` George Janelidze
[not found] ` <E258C2F0-8620-4CD8-8011-B544D44C95BD@dima.unige.it>
[not found] ` <027601ca62f9$ace359c0$0b00000a@C3>
2009-11-12 8:12 ` Question on exact sequence (by G.J.) Marco Grandis
-- strict thread matches above, loose matches on Subject: below --
2009-11-13 2:05 Question on exact sequence F William Lawvere
2009-11-12 19:58 Michael Barr
2009-11-11 17:29 Marco Grandis
2009-11-11 17:15 Marco Grandis
2009-11-11 16:34 Clemens.BERGER
2009-11-11 15:04 George Janelidze
2009-11-12 12:41 ` Michael Barr
2009-11-13 16:06 ` Michael Barr
[not found] ` <00a001ca63f6$80936b50$0b00000a@C3>
[not found] ` <Pine.LNX.4.64.0911122132300.27416@msr03.math.mcgill.ca>
[not found] ` <000f01ca644d$065eb590$0b00000a@C3>
[not found] ` <Pine.LNX.4.64.0911131101330.27416@msr03.math.mcgill.ca>
2009-11-13 18:15 ` George Janelidze
2009-11-14 16:24 ` Michael Barr
2009-11-15 14:35 ` George Janelidze
2009-11-16 16:43 ` Marco Grandis
2009-11-13 0:16 ` George Janelidze
2009-11-10 20:14 Ross Street
2009-11-10 16:15 Michael Barr
2009-11-10 14:44 Marco Grandis
2009-11-10 3:22 Steve Lack
2009-11-09 22:57 Michael Barr
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