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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!)

flag	asked Apr 13 2011 at 22:58 Qfwfq 8,640●35●86

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See perhaps this answer of Urs Schreiber: mathoverflow.net/questions/6125/… – Kevin Lin Apr 14 2011 at 1:21

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Also this: mathoverflow.net/questions/4214/… – Kevin Lin Apr 14 2011 at 1:22

wow! thanks for this link! – SGP Apr 14 2011 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 2011 at 12:20

SGP · Answer 1 · 2011-04-13 23:14:28Z UTC

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).

Is there a common general setup for both Weil cohomologies and generalized cohomology theories?

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