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My question can be simply (and loosely) stated as follows:

Is there a general (but not too general) construction that captures, as specializations, both Weil cohomologies in algebraic geometry and generalized cohomology theories in topology ?

(I must say I'm not an expert of any of the two!)

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See perhaps this answer of Urs Schreiber:… – Kevin H. Lin Apr 14 '11 at 1:21
Also this:… – Kevin H. Lin Apr 14 '11 at 1:22
wow! thanks for this link! – SGP Apr 14 '11 at 2:37
It seems I didn't notice that this question had already been asked. Thank you for the links to the answers and nLab. – Qfwfq Apr 14 '11 at 12:20

It would appear that the answer to the question as stated is no. However, Voevodsky's motivic homotopy theory does provide an adequate framework for both Weil cohomologies and generalized cohomology theories; there is a version of "Brown representability" theorem (representing object for a generalized cohomology theory) which is exploited in the applications to K-theory (Milnor conjecture and Bloch-Kato conjecture).

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