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There is defined several cohomology theories for algebraic varieties, but in the situation is very different for topological spaces (up to homotopy) for which there is only one cohomology theory for every given abelian group of coefficients.

What is really the difference? Can one define a similar category of motives of topological spaces? If so what is the property of this abelian tensor category which implies the uniqueness of fiber functors?

Thanks!

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    $\begingroup$ The situation is more complicated than you describe for topological spaces. There are extraordinary cohomology theory that are not uniquely determined by the cohomology of a singleton. I don't know enough about the cohomology of algebraic varieties / motives to answer your question, though. $\endgroup$ – Najib Idrissi Jan 19 '15 at 8:33
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    $\begingroup$ The comparison is not entirely fair. For instance, the cohomology of quasicoherent sheaves on varieties should not be compared to, say, singular cohomology with constant coefficients but rather a cohomology theory with local coefficients (e.g. sheaf cohomology). $\endgroup$ – Zhen Lin Jan 19 '15 at 8:35
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    $\begingroup$ Extraordinary cohomology theories are without the triviality assumption on the cohomology of a point, which I assume here as one of axioms. $\endgroup$ – Mostafa Jan 19 '15 at 8:36
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First, it is not quite true that there is "only one cohomology theory" for topological spaces. One should rather say that there are several such theories, but that for reasonable spaces, they give the same results.

The reason why this is true in a category of reasonable topological spaces is that nice topological spaces can be "built up" from smaller ones in various ways, (for instance, see the definition of a CW complex). From the Eilenberg-Steenrod axioms, one can formally calculate the cohomology of an interval, and then a circle, and then a sphere, and so on. The Eilenberg-Steenrod axioms are sufficient to calculate the cohomology of any space which is "built up" from the point using a few basic operations.

In the world of algebraic geometry, this is pretty much impossible. Algebraic varieties cannot in general be decomposed into simpler algebraic varieties by any process. The situation is much more "rigid". For instance, there is no way to deduce the cohomology of an algebraic curve from the cohomology of a point using simple axioms, despite algebraic curves being the simplest objects after the point.

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A big difference between topological spaces and varieties is that, for topological spaces, we can define $H^{\ast}(X, \mathbb{Z})$ where as, for varieties, there is no cohomology theory with coefficients in $\mathbb{Z}$. (I've blogged about this here and here; the examples I'm giving are originally due to Serre.) This is important because, in the topological setting, we have the universal coefficient theorem which says that $H^{\ast}(X, A)$, for any ring $A$, can be computed from $H^{\ast}(X, \mathbb{Z})$. For varieties, there is no universal theory to play the same role. (Although, as I imagine you know, motives are an attempt to come as close as possible.)

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