Magic Motives II
If sets and vector spaces are secretly the same thing, and spaces are built out of these new sets, then there should be one big category of spaces that has everything in it, including invariants!
For instance, a vector space of dimension $n$ over a finite field may be represented by all $n$-tuplets of MUB operators representing the field. Now these tuplets are just collections of arrows in the category of fancy sets. A triplet of $3 \times 3$ matrices with cubed roots of unity would act on a $27$ element set, via Cartesian product. The collection of all finite dimensional vector spaces over $F_3$ would live in the set subcategory made up of Cartesian powers of the three element set.
Similarly, spaces for the two element field $F_2$ may occupy powers of the two element set, usually denoted by $\Omega$ in the ordinary topos. This suggests that a power set monad for fancy sets might have something to do with invariant functors for $F_2$ (or even $2$-adic numbers). How cool would that be? The world of motives would then return to Grothendieck's dream.
For instance, a vector space of dimension $n$ over a finite field may be represented by all $n$-tuplets of MUB operators representing the field. Now these tuplets are just collections of arrows in the category of fancy sets. A triplet of $3 \times 3$ matrices with cubed roots of unity would act on a $27$ element set, via Cartesian product. The collection of all finite dimensional vector spaces over $F_3$ would live in the set subcategory made up of Cartesian powers of the three element set.
Similarly, spaces for the two element field $F_2$ may occupy powers of the two element set, usually denoted by $\Omega$ in the ordinary topos. This suggests that a power set monad for fancy sets might have something to do with invariant functors for $F_2$ (or even $2$-adic numbers). How cool would that be? The world of motives would then return to Grothendieck's dream.
posted by Kea | 2:08 PM
1 Comments:
Gradually I pick up little details. For instance, I can see why it is that I like (Pauli) density matrices; they live in a space that supports matrices, vectors, scalars, complex numbers, etc. But this is basic Clifford algebra.
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