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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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    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.2012-12-12

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