Motif of Motives II

Carl Brannen has reminded me of Cartier’s classic paper, A Mad Day’s Work. He discusses everything, from Grothendieck’s biography to symmetry groups for a point. In particular, he points out that a sensible notion of symmetry group for a point comes from considering points as functors between toposes. Since there are natural transformations between functors, one might find a group of invertible natural transformations between a functor and itself.

The really cool thing about all this is that the group is not fundamental. Eat your heart out Gauge Theory!

Which reminds me that I meant to say something about Grothendieck’s motives. As Cartier explains, motives are a part of Grothendieck’s dream, a vision of unifying number theory and modern topology, and hence almost everything else as well. The theory of motives is still mysterious, although an impressive amount of progress in the related physics and mathematics has been made in the last 30 years. Consider for example the work of Kontsevich on motives and operads in deformation quantization. It’s kind of funny that the mathematicians have chosen a word (motives) that starts with M. It’s their version of M-theory!

An important intuition behind motives is that of projective geometry. Motives obey powerful relations, an example of which is the equation

M(projective plane) = M(plane) + M(line) + M(point)

which expresses the usual grading of a projective plane (over any field) into an affine space with a line and point at infinity. This feature of a grading in dimension is typical of motives, as it is for categorical dimension.

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