e-mail to martin brandenburg

hi. i've started reading the math overflow discussion, and i can see now that there are a lot of issues to discuss. so many that i'm not quite sure where to start...

also i noticed suggestions about carrying out our discussion in a place where other people can follow it, which probably seems like a reasonable idea, though i'm not sure of the best way to implement it...

also i've already tried to warn about my slowness at writing. i should emphasize this again. this slowness is making me wonder whether it might make sense to try to carry out some of our discussion via speaking (over skype, for example). of course this might conflict to some extent with the previous suggestion to carry out the discussion in a place where others can follow it (presumably by
reading)...

let me start by trying to describe perhaps the simplest version of an equivalence between the concepts of "dimensional category" and of "commutative algebra graded by an abelian group" ("graded commutative algebra" for short). in this version we get an equivalence of 1-categories from the category of graded commutative algebras to the category where an object is a "strict" dimensional category and a morphism is a "strict" dimensional functor.

(there's a tradeoff between the usefulness of omitting the strictness requirements in many contexts, and the complication of dealing with an equivalence of 2-categories rather than of 1-categories.)

"strict" here can be taken to mean that the equations that might be weakened to natural isomorphisms (associativity and commutativity of tensor product and of the unit and inverse laws for tensor product in dimensional categories, and preservation of tensor product and unit objects and inverse objects by dimensional functors) are actually required to be equations.

the equivalence is now given (as ben webster indicated) by taking the objects in the dimensional category to be the grades in the graded commutative algebra and the elements in the hom-space [x,y] to be the elements in the grade y-x, and the inverse equivalence by taking the grades to be the objects and the elements in x to be the elements in [1,x].

(i hope that describing the inverse equivalence as above might be enough to make the details clear, but if not then i can try to give them more explicitly.)

this construction gives an equivalence from "graded commutative monoids in x" to "strict x-enriched dimensional categories" under fairly mild conditions on the symmetric monoidal category x. for example when x is the category of sets with tensor product given by cartesian product, a graded commutative monoid in x is a commutative monoid equipped with a homomorphism to an abelian group, and an x-enriched dimensional theory is a "toric dimensional theory".

by the way, notice that the dimensional category corresponding to a graded commutative algebra x can be non-equivalent (even as just a plain category) to the dimensional category of invertible graded modules of x (for example if x is an ungraded commutative algebra with non-trivial ideal class group).

i'll stop here for now, in the interest of moving the discussion along and trying not to get bogged down by my slowness of writing. i'm eager to resolve questions about the universal properties of tensor categories of coherent sheaves (and so forth), in part because i often find myself working in isolation (exacerbated by my difficulties in understanding many parts of the standard literature) and wind up
reinventing (with roughened edges aka lots of mistakes) things that are already known; i hope that this discussion helps me in understanding many things that are already known.