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Weibel defines an abelian subcategory of an abelian category $A$ to be a subcategory $B$, which is an abelian category, such that a sequence of two maps in $B$ of is short exact iff it is short exact in $A$.

Does someone know of a concrete example of 2 abelian categories, one a subcategory of the other, but not an abelian subcategory?

Thanks.

Elle Najt
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    Closely related (and in fact, my answer includes an answer to your question): http://math.stackexchange.com/questions/1441395/is-a-kernel-in-a-full-additive-subcategory-also-a-kernel-in-the-ambient-abelian – Eric Wofsey Oct 23 '15 at 01:53

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Take $A$ to be the category of presheaves of abelian groups on some space, and take $B$ to be the subcategory of sheaves of abelian groups. The inclusion functor $B \to A$ is a right adjoint, so preserves limits, but it generally does not preserve colimits, such as cokernels, and in fact it is not exact. Hence most short exact sequences of sheaves fail to be short exact sequences of presheaves.

Qiaochu Yuan
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