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One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi_1(X_\overline K)$, where $X$ is a (anabelian) variety and $X_\overline K=X\times_K \overline K$. This yields a representation by outer automorphims of the fundamental group.

Having been recently exposed to some of the ideas of Grothendieck in his Esquisse and letter to Faltings that this representation seems to be quite a natural action, given the definition, it seems to beg the (perhaps naive) question: how have the ideas of anabelian geometry, or homotopical algebra in general, been applied to the study of Galois representations?

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