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I have a question about the definition of $\ell$-adic local systems. I understand how to define local systems over any finite extension of $\mathbb{Q}_{\ell}$, but not how to take the "union" of these categories to obtain the category of $\overline{\mathbb{Q}}_{\ell}$-local systems. Any reference or explanation of this construction would be appreciated.

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If I had to guess, I would say that it is obtained by taking the inverse limit of all the categories of $K$-local systems, as $K$ varies over the finite extensions of $\mathbb{Q}_\ell$. But that is just a guess. –  Zhen Lin Dec 12 '12 at 8:18
    
Shouldn't it be a direct limit instead? Also, I have no idea how to take a (co)limit of categories. –  Justin Campbell Dec 12 '12 at 17:41
    
If $K$ is an extension of $L$, we get an obvious functor from $L$-modules to $K$-modules. Of course there's one in the other direction as well, but that's more difficult. –  Zhen Lin Dec 12 '12 at 18:11
    
What is the obvious functor? The construction of $L$-local systems for $L/\mathbb{Q}_{\ell}$ finite is already pretty complicated. I think there should be something like this and we should take the "direct limit" over the resulting diagram of categories, but this still leaves open what these "extension of scalars" functors are and what objects and morphisms look like in the colimit category. –  Justin Campbell Dec 12 '12 at 18:17
    
Agh, typo. If $L$ is an extension of $K$ then there is an obvious functor from $L$-modules to $K$-modules (restriction of scalars). So the fact that $\overline{K}$ is the directed colimit of its finite subextensions gives us an inverse diagram of module categories. I imagine something similar works here. –  Zhen Lin Dec 12 '12 at 21:53

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This construction can be found in Deligne's "La Conjecture de Weil II." Deligne calls it a "2-limite inductive."

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