proj geom = dimensional analysis 6 (outline)

1 .... preamble and introduce baez's example ....

2 introduce two ways of formally interpreting the example, namely graded commutative algebra way and (more interestingly) lawvere's way, as "dimensional category" ....

(?? peculiar aspect here ... not really focusing on (?? contrast/relationship between 2 sides of) title equation .... rather, on contrast/relationship between standard approach and lawvere approach ... ?? towards .... understanding projective geometry ???? ..... ???? ...... towards establishing title equation ???? ..... .... some confusion here ... ???? ......)

3 using both ways from 2 above to connect to projective geometry .... _much_ more interestingly using lawvere's way ... ??? "now there are at least two ways to formalize what we're doing here in such a way as to reveal the connection to projective geometry" ...... ?????? ...... "first traditional way with perhaps most obvious direct connection to nominal topic of this conference; but more interestingly, ..." ..... ???? .....

3.1 ...."proj" ... "modding out by scaling group" ..... ???? ..... ????? ........ .... ??? "... try to return to this later ..."

3.2 ..... "taking moduli stack of models of dimensional theory" .... leading to 4 .... ????sa "model as state of affairs" .... "theory with free parameters...." .... ???? .....

4 rant about lawvere's concept of "theory" ... ??? and beck's concept of "doctrine" .... ??? and these as forming one approach ("categorical logic" approach ???? .....) to formulating/understanding/developing "tannakian program" / "geometric function theory" ...... ??????? ..... ???? include stuff (real rant ...) about .... word "theory" .... ideas it's supposed to encompass .... logic(tie in with mention of "categorical logic" above), physics .... ???? .... ??? also include stuff about higher doctrines and vector bundles and quasicoherent sheafs, going beyond line bundles .... ???? .... ???? lawvere's idea of theory as cat .... tied in with stuff, structure, propety = type, predicate, axiom .... ???? ..... ??gauge theories and general relativity as prototypical of physical theories where state of affairs may have automorphism / moduli stack may be genuinely stacky ... ?? ...

???? where in above (?? and/or below??) do we give the two basic dictionary entries (dimension = line bundle, quantity of given dimension = section of given line bundle) ??? ..... ???? should be semi-early, not too late ???? .....

5 ?? return as promised in 3.1 ??? ..... orbit stack .... "unification" approach to tammakian program .... ??? ....

??? somewhere try to work in "walking line-nicely-embedded-in-standard-n-space" .... brandenburg's correction .... ??? ..... ?? tied in with emphasis on "algebraic geometry as part/aspect of categorical logic" .... ???? ....

??? have we managed above yet to work in all the various approaches to tannakian program that we want to ?? ...

?? maybe part of some sort of cocnclusion .... "algebraic geometry for category theorists" idea ... that is, not to trust algebraic geometers's ideas as to how to apply category theory in algebraic geometry .... ??? ... instead, philosophy of "categorical loguc" .... ???? ....