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So this is a crazy question, but I will try to give at least a partial answer. This question about the Beilinson regulator is also relevant, and this is also an attempt to reply to the comments of Ilya there. I apologize for simplifying and glossing over some details, see the references for the full story.

So this is a crazy question, but I will try to give at least a partial answer. This question about the Beilinson regulator is also relevant, and this is also an attempt to reply to the comments of Ilya there. I apologize for simplifying and glossing over some details, see the references for the full story.

The third key point is that Weil cohomology theories are always "ordinary" in some sense, i.e. in some framework of oriented cohomology theories they would correspond to the additive formal group law (see Lurie: Survey on elliptic cohomology). If we allowed more general (oriented) cohomology theories, the universal cohomology would not be pure motives, but algebraic cobordism.

The third key point is that Weil cohomology theories are always "ordinary" in some sense, i.e. in some framework of oriented cohomology theories they would correspond to the additive formal group law (see Lurie: Survey on elliptic cohomology). If we allowed more general (oriented) cohomology theories, the universal cohomology would not be pure motives, but algebraic cobordism.

There are (at least) three key points to mention here: one is that a Weil cohomologies are "geometric" theories, as opposed to "absolute". For example, when considering etale cohomology, we are considering the functor given by base changing the variety to the absolute closure of the ground field, and then taking sheaf cohomology with respect to the constant sheaf Z/l for some prime l, in the etale topology. The "absolute" theory here would be the same, but without base changing in the beginning. In the classical literature, and in number theory, the geometric version is the most important, partly because it carries an action of the Galois group of the base field, and hence gives rise to Galois representations. On the other hand, the absolute version is important for example in the work of Rost and Voevodsky on the Bloch-Kato conjecture. Similarly, it seems like cohomology theories in general come in geometric/absolute pairs.

It seems like all the cohomology functors one typically considers can be defined not only for smooth projective varieties, but also for more general varieties. One could again hope for a category which has a similar universal property as above, but now with respect to all varieties. This category would be the category of mixed motives, and in the standard conjectural framework, one hopes that it should be an abelian category. It is not clear whether this category exists or not, but see Levine's survey above for a discussion of some attempts to construct it, by Nori and others. If we had such a category, a suitable universal property would imply that there are realization functors again, now to "mixed" categories, for example mixed Hodge structures. The realization functors would induce maps on Ext groups, and a suitable such map would be the Beilinson regulator, from some Ext groups in the category of mixed motives (i.e. motivic cohomology groups) to the some Ext groups which can be identified with Deligne-Beilinson cohomology.

There are (at least) three key points to mention here: one is that a Weil cohomologies are "geometric" theories, as opposed to "absolute". For example, when considering etale cohomology, we are considering the functor given by base changing the variety to the absolute closure of the ground field, and then taking sheaf cohomology with respect to the constant sheaf Z/l for some prime l, in the etale topology. The "absolute" theory here would be the same, but without base changing in the beginning. In the classical literature, and in number theory, the geometric version is the most important, partly because it carries an action of the Galois group of the base field, and hence gives rise to Galois representations. On the other hand, the absolute version is important for example in the work of Rost and Voevodsky on the Bloch-Kato conjecture, and in comparison theorems with motivic cohomology. Similarly, it seems like cohomology theories in general come in geometric/absolute pairs.

It seems like all the cohomology functors one typically considers can be defined not only for smooth projective varieties, but also for more general varieties. The right notion of cohomology here seems to be axiomatized by some version of the Bloch-Ogus axioms. One could again hope for a category which has a similar universal property as above, but now with respect to all varieties. This category would be the category of mixed motives, and in the standard conjectural framework, one hopes that it should be an abelian category. It is not clear whether this category exists or not, but see Levine's survey above for a discussion of some attempts to construct it, by Nori and others. If we had such a category, a suitable universal property would imply that there are realization functors again, now to "mixed" categories, for example mixed Hodge structures. The realization functors would induce maps on Ext groups, and a suitable such map would be the Beilinson regulator, from some Ext groups in the category of mixed motives (i.e. motivic cohomology groups) to the some Ext groups which can be identified with Deligne-Beilinson cohomology.