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I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard examples thereof, are essentially, or are supposed to be, generalizations or algebraic versions of singular cohomology. If I am incorrect in this assessment, please do correct me.

Meanwhile, we have other interesting cohomology theories in topology: for example (topological) K-theory, elliptic cohomology, complex cobordism, .... Correspondingly, then, are there notions of "K-Weil cohomology theory" or "elliptic Weil cohomology theory", etc.? Is it possible?

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Yes, it is quite possible. Keywords for this kind of technique in homotopy theory include motivic (stable) homotopy theory and $\mathbb{A}^1$-homotopy theory. Algebraic K-theory makes its appearance as an analogue of topological K-theory. There is also a Landweber exact functor theorem which can produce some elliptic cohomology theories (but not a universal one). If you come up with something interesting about motivic elliptic cohomology theories then you should write a paper about it. –  Tyler Lawson Jun 5 '10 at 1:52
    
Why is there no universal motivic elliptic cohomology theory? Do you mean there can't be or that such a thing might exist but hasn't been found yet? –  Chris Brav Jun 5 '10 at 2:08
    
There is a motivic version of the Landweber exact functor theorem (the version I know is due to Naumann-Spitzweck-Østvaer) and this suffices to construct many elliptic cohomology theories, including a "universal elliptic cohomology" away from the primes 2 and 3 in the same manner that it is definable in the old sense. However, at the primes 2 and 3 the fact that the LEFT is only functorial on the homotopy category means together with the fact that certain elliptic curves have 2- and 3-primary automorphism groups means a universal theory is difficult, motivic or not. Motivic is always harder. –  Tyler Lawson Jun 5 '10 at 4:21
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