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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: mathoverflow.net/questions/6125/… –  Kevin H. Lin Apr 14 '11 at 1:21
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Also this: mathoverflow.net/questions/4214/… –  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
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1 Answer

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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