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I posted the following question on MathStackexchange, where it was suggested that I should move my question to Mathoverflow, which do here (https://math.stackexchange.com/questions/3550741/algebraic-geometric-analogue-of-browns-representability). In topology, the Brown representability theorem is very useful to show for instance that the functor $X\rightarrow H^i(X,A)$ is representable. My question is whether or not there exists an algebraic geometric analogue of the (pointed) homotopy category of CW complexes and of Brown's representability theorem. I would like to use this to show the existence of an $\ell$-adic analogue of the Eilenberg-MacLane space, i.e. to show the functor $$X\rightarrow H^i_{ét}(X,\mathbb{Q}_{\ell})$$ is representable.

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