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Is there an example that the category of sheaves of abelian groups on a site is not an abelian category?

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Sheaves of what? –  Qiaochu Yuan Feb 12 '10 at 6:15
Abelian groups, sorry for the unclearness –  Taisong Jing Feb 12 '10 at 6:35
The Wikipedia article on abelian categories says that all such categories are abelian. –  Qiaochu Yuan Feb 12 '10 at 6:39
I see, thanks a lot! –  Taisong Jing Feb 12 '10 at 6:50

1 Answer 1

The category of sheaves of abelian groups on a site is always abelian. See Theorem 2.1.4 on page 20 of

  • Michael Artin - Grothendieck Topologies (1962)

which is available online: http://www.math.ubc.ca/~gor/Artin-GT.pdf

Once you construct the sheafification functor in the context of an arbitrary site most of the basic results in the classical theory of sheaves can be carried over to the context of sites without (much) modification.

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I would argue that this suggestion has the wrong flavor to it. First, it is not true in general that there exists a sheafification functor for sheaves taking values in an arbitrary category. This is known to fail in almost every case. It is known, however, that there is always a sheafification for sheaves of sets. Then, instead of saying that we have a sheaf of abelian groups, we really have an abelian group in the category of sheaves of sets. We can see that the category of abelian groups in a Grothendieck topos is necessarily abelian with just a little bit more effort. –  Harry Gindi Feb 17 '11 at 21:38
But sheafification always exists for sheaves taking values in the category of abelian groups. It seems odd to me to disregard sheafification simply because it doesn't work for arbitrarily-valued sheaves. All of the intuition from the classical theory of abelian sheaves carries over to the context of sites by thinking in this way. Other than sheaves of sets, how often do we come across sheaves with values is a non-abelian category? –  Beren Sanders Feb 17 '11 at 22:14
@Beren: You'd be surprised what computer-scientists and logicians do with sheaves. –  Andrej Bauer Feb 17 '11 at 22:18
Although it is nice that we can think about sheaves of abelian groups as abelian group objects in the category of sheaves of sets (the Grothendieck topos Harry refers to), personally this isn't the most transparent way for me to understand why the category of abelian sheaves on a site in abelian. –  Beren Sanders Feb 17 '11 at 22:20
I edited, because it is simply not true that sheaves on an arbitrary site form an abelian category. Only sheaves of abelian groups. You may argue this is common usage in algebraic geometry, but not in category theory, and someone stumbling across this answer may get in a muddle. As to sheaves valued in other categories, what about sheaves of groupoids? Sheaves of crossed modules? Sheaves of arbitrary groups? e.g. arxiv.org/abs/0909.3350 –  David Roberts Feb 17 '11 at 22:36

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