Holmstrom Some ideas III

Category theory

The construction of an L-function from a motive is not a functor, but rather something like a “ring HM” from a “tensor category” to a certain ring of complex-valued functions, probably a C-algebra. Can we study such “ring HMs” more generally and systematically??? Perhaps the Brauer group is such a structure. What about Poincaré series? Perhaps something is in Joyce. In general, there should be many interesting examples of structure-preserving maps from Cat to Set. For example, constructions preserving limits (i.e. a “continuous” construction). More general slogan: anything you can do with sets, you can do with (higher) cats.

To what extent can structure on Hom groups be viewed as “coming from the representing object”. Example: A representable setvalued functor on schemes factors through the forgetful functor from Grp iff the representing object is a group scheme. A cohomology theory in Algebraic Topology has a multiplicative structure iff the spectrum is a ring spectrum. For any functor $F: C \to D$, and any (forgetful?) functor $S \to D$, we say that $F$ is “of category $S$ type” if it factors through $S$. Restricting attention to representable functors give a definition of a category $S$ object in $C$.

Q: How is this related to the concept of a “spectrum object” in a category? And why does it seem like you cannot define a “ring object” or “R-module object” in a category by the diagram method?

Endomorphisms of cohomology theories

The concept of cohomology operation should fit into a systematic study of the endomorphisms of a cohomology theory. For an example of the fruitfulness of this viewpoint, look at Nori’s construction of mixed motives: Roughly speaking, a mixed motive is a module over the ring of endomorphisms of the singular cohomology functor on pairs of schemes.

A fundamental question is this: Is the functor from spectra to cohomology functors full and faithful?

Regulators

For regulators, see also Hannu email. Ostvaer said something about regulators and his project with Barwick.

Iwasawa theory

See Mitchell survey. Is it possible/interesting to develop elliptic cohomology of varieties?

Rigid analytic geometry

Stable homotopy of rigid analytic geometry. See Riou machinery, K-theory 0825

nLab page on Some ideas III