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Taking direct limits is an exact functor in the category of modules. It has been discussed extensively here.

What I ask is: I know that taking direct limits is not an exact functor in other categories. Our professor mentioned it when discussing Cech cohomology of sheaves, describing the n-th Cech cohomology group as a direct limit. (I can provide details if it's necessary)

Can you please help me find an example in some category where taking direct limits is not an exact functor? Categories of groups or topological spaces maybe?

Thank you.

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Neither the category of groups nor the category of topological spaces is additive. How would you even define exactness for the direct limit functor for these categories? –  Rasmus Mar 23 '12 at 9:36
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@Rasmus: a general definition is that a functor between two categories with finite limits is exact if it preserves finite limits, and this specializes to the usual definition for abelian categories. –  Qiaochu Yuan Mar 23 '12 at 9:44
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You probably want to clarify that by direct limit, you mean a directed limit or filtered colimit. (In some circles, direct limit is just a synonym for colimit.) –  Zhen Lin Mar 23 '12 at 11:34
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1 Answer

up vote 1 down vote accepted

Consider the opposite category of the category of Abelian groups. In that category, direct limits will not preserve exact sequences because in the category of Abelian groups inverse limits don't preserve exact sequences.

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Thanks for the answer, but could you please provide a sketch of proof, or at least a bit more detailed explanation why doesn't inverse limit preserve exactness in $\bf{Ab}$? Also, a concrete example would be good. Thank you. –  AdrianM Mar 23 '12 at 10:00
    
@AdrianM See here. Most books on homological algebra discuss this issue. –  Rasmus Mar 23 '12 at 10:30
    
Ok, got it, thanks. Any example from (somehow elementary) sheaf theory? –  AdrianM Mar 23 '12 at 11:57
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