# What's the difference between cohomology theories of varieties and topological spaces

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!

• 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. Jan 19 '15 at 8:33
• 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). Jan 19 '15 at 8:35
• Extraordinary cohomology theories are without the triviality assumption on the cohomology of a point, which I assume here as one of axioms. Jan 19 '15 at 8:36

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.)