Talk:Exact sequence/Archive 1

Image

 * A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms.  More generally, the notion of an exact sequence makes sense in any abelian category (i.e. any category with kernels and cokernels).

Why can't we just use the category theoretic definition of image. Isn't this more general? In fact doesn't it subsume all the instances? --174.119.186.126 (talk) 01:31, 14 September 2010 (UTC)

0 or 1 to denote the trivial group
There was a sentence that it is customary ot use 1 to denote the trivial group rather than 0. This does not agree with the notation used in the rest of the article nor have I ever seen an exact sequence written as $$ 1\rightarrow A\rightarrow B\rightarrow B/A \rightarrow 1 $$. Maybe thats because the exact sequences I've mostly come across are from the category of modules over a ring which is abelian (whereas nonabelian groups can be written multiplicatively). But anyways, it seemed confusing that the rest of the article denotes trivial groups as 0 so I just removed that sentence. LkNsngth (talk) 21:20, 10 August 2009 (UTC)


 * I had noticed the same thing independently just now, and added a few words to cover at least the case of non-abelian groups.Daqu (talk) 15:19, 22 October 2010 (UTC)

Abelian only?
This article seems to be assuming all exact sequences are between Abelian groups. What about non-Abelian ones - eg the inclusion of any normal subgroup followed by the projection to the quotient, or the long exact sequence in homotopy which involves both Abelian and (in general) non-Abelian groups? Simplifix (talk) 23:04, 1 February 2011 (UTC)

equalizers
As pointed out on the talk page to sheaf (mathematics), it is often the case that one of the arrows is an equalizer, i.e. there are also two parallel arrows, and that this is how the Mayer-Vietoris sequence is constructed. It would be nice if some kind of explicit discussion of this case was handled here. linas (talk) 21:57, 18 August 2012 (UTC)
 * After some digging, it appears that the coequalizer article provides the needed statement that its a generalization of the idea of a quotient. Then, in the examples section, it even gives a the standard homological example of gluing two arcs together to make S^1. Yay! What we need now is to transpose all of that into this article... linas (talk) 23:02, 18 August 2012 (UTC)