Talk:Exact sequence

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[edit] 2Z --> Z or Z -->2Z

there were a couple of corrections by anonymous editors recently that I've just reverted. There seem to be three different choices for the example exact sequence:

  1. 0 → Z → Z → Z/2Z → 0
  2. 0 → Z → 2Z → Z/2Z → 0
  3. 0 → 2Z → Z → Z/2Z → 0

The first two are pretty much the same, the second map is n to 2n, and the only question is how you want to label it. The third one is slightly different, the second arrow is an inclusion map. The anonymous editors have gone through all three, and I reverted back to the original, which is #1. But actually, I prefer #3, because it shows more explicitly the general paradigm that for any quotient group B/A, you have an exact sequence 1 → A → B → B/A → 1, whereas the other sequences don't have the names in the right places. I wonder what others think. -lethe talk 01:26, 27 January 2006 (UTC)

Actually, I think the second one is wrong: the image of 2Z → Z/2Z is 0, while the kernel of Z/2Z → 0 is {0,1}, so that's not exact. -lethe talk 07:05, 27 January 2006 (UTC)

I prefer the first because it keeps the external diagram external. 2Z makes sense as the kernel in the quotient Z/2Z, but is uneccessary if not confusing as the second group in #3. MotherFunctor 06:01, 14 May 2006 (UTC)

I'm not sure what you mean by "external diagram". Can you explain? Cute handle by the way. -lethe talk + 06:42, 14 May 2006 (UTC)
Thanks and sure. It comes from a nice categorical set theory book "Sets For Mathematics" Lawvere, Rosenbrugh. External diagram labels objects and arrows, internal diagram shows behavior of arrows on points in object. \mathbb Z and \mathbb Z/\mathbb Z_2 are objects. n\mathbb Z is not an object, unless it's another name for \mathbb Z. Anyway, I think it is bad style, as is evident from the confusion. The first one is nice. MotherFunctor 05:46, 17 May 2006 (UTC)