Magic Motives

The most fashionable of all phrases amongst real mathematicians studying QFT, occasionally mentioned here, are the words motivic cohomology. This is supposed to be the mother of all cohomology theories. But what is cohomology anyway?

As a basic concept, cohomology relies on the logic of ordinary sets, particularly the difference between unions and disjoint unions. The latter type of union keeps track of which set each element actually came from, so it tends to double count intersections. In the category Set, disjoint union is a coproduct. Ordinary unions are a little less natural, but intersections are pullbacks of subset monics.

Universality is a basic categorical concept. A cohomology theory usually assumes a standard set of axioms for a cohomology functor $H^{*}$ from some category of (commutative) spaces (which we want invariants for) to a category of groups (where invariants live). The existence of group inverses is related to the arbitrariness of path directions in a commutative space. A universal cohomology theory is supposed be about an elusive category of motives.

But thinking of the field with one element, what happens if we have sets with funny actions, instead of just sets? Are our spaces necessarily commutative? Perhaps the underlying logic of intersection and union should be modified to properly account for the geometry of the absolute point. This might require drawing higher dimensional limit diagrams, or looking at higher dimensional target categories, not necessarily groupoids. Fortunately, some really smart people are now thinking about noncommutative motives, but it may take a while for us poor physicists to see what is going on.