2 added 28 characters in body

After reading the answer to my previous question: http://mathoverflow.net/questions/71587/what-are-the-different-theories-that-the-motivic-fundamental-group-attempts-to-un

I decided to read up on Tannakian formalism.

Given the category of numerical motives, and assuming Conjecture C of the standard conjectures (the one regarding the grading of numerical motives), one can construct a category that will be Tannakian. This will be done by changing the sign of the canonical'' morphism $h^iX\otimes h^jX \cong h^jX \otimes h^iX$ for $ij$ odd .

It seems in texts about motives, that the end goal was always to achieve a Tannakian category. But what motivation is there for this? Why would a category that has to do with motives be the category of representations of an affine group scheme? This seems crazy to me. Is this immitative of some easier, more well-understood, theory in which it make sense to relate cohomology with representations?

Also, is it conjectured what this mysterious affine group scheme is, in the case of numerical motives with the adjustment written above?

1

# Why would the category of Motives be Tannakian?

After reading the answer to my previous question: http://mathoverflow.net/questions/71587/what-are-the-different-theories-that-the-motivic-fundamental-group-attempts-to-un

I decided to read up on Tannakian formalism.

Given the category of numerical motives, and assuming Conjecture C (the one regarding the grading of numerical motives), one can construct a category that will be Tannakian. This will be done by changing the sign of the canonical'' morphism $h^iX\otimes h^jX \cong h^jX \otimes h^iX$ for $ij$ odd .

It seems in texts about motives, that the end goal was always to achieve a Tannakian category. But what motivation is there for this? Why would a category that has to do with motives be the category of representations of an affine group scheme? This seems crazy to me. Is this immitative of some easier, more well-understood, theory in which it make sense to relate cohomology with representations?

Also, is it conjectured what this mysterious affine group scheme is, in the case of numerical motives with the adjustment written above?