??so consider for example tag theory given by pairs of N-sets, vs one given by N^2-sets ... and so forth .... ??something about evident lack of favorite model in former case ... and so forth ...

??also something about toric dimensional theory and plain dimensional theory, and so forth ... nature (??something about "stucture on" and so forth ...) of moduli stack of models ... and so forth ...

??something about ... ???? dim th : enveloping ag th :: abelian group : its group algebra ... ???something about "group algebra" functor (valued in commutative algebras...) as preserving algebraic (weak...) colimits but not algebraic limits ... ???? thus geometric limits but not geometric colimits ... ??? ???hmm ...??seems slightly strange, since... ??i associate "dimensional theories" with "projective geometry" (more or less...) and i think of projective geometry as (of course ...??...) lying on the "good" side as far as geometric colimits are concerned ...

so then what _about_ the geometric interpretation of algebraic products of dimensional theories??? .... and so forth ...

???let's try... the algebraic product of [the usual dimensional theory of the projective line] with itself ...

... 0 0 0 *1* 2 3 4 5 ...

2 3 4 5 6 7 ....
3 4 5 6 7 8 ....
4 5 6 7 8 9 ....
5 6 7 8 9 10....
6 7 8 9 10 11....
.
.
.

??so then what about how graded modules of the above compare to pairs of graded modules.... ????... and so forth ...

??well, so what _is_ this the dimensional theory of ??? ... ??an idempotent number, together with a pair of line objects ...


???what about something about ... ???over P^1 + P^1, taking the line bundles given by "dual tautological over the first line and trivial over the second" and "trivial over the first and dual tautological over the second" ???...

???"an idempotent number z, together with a pair x,y of line objects and sections x#,x1,x2,y#,y1,y2 such that when z=0 then x#=0 and y# is invertible and y1=y2=0 whereas when z=1 then x# is invertible and x1=x2=0 and y#=0" ??? or something??? ???does that actually make any sense??? ... and so forth ...

??zx=x ... z*y1=y1 , z*y2=y2 ... (1-z)y=y , (1-z)x1=x1 , (1-z)x2=x2 ...

??but what about the idea of trying to get things to be invertible here??? ...??does that make any sense ?????.....

??hmm, maybe we left out some generating sections ... ???instead of x#, how about x+ in grade x and x- in grade -x ??... and so forth ... ???.... ???but does that screw up the numerology ??? ...??? ...

???hmmmmm..... ?????..... ??or maybe un-[screw-up] it???...

0 0 0 1 2 3 ....

0 0 0 0 1 2 3
0 0 0 0 1 2 3
0 0 0 0 1 2 3
1 1 1 1 2 3 4
2 2 2 2 3 4 5
3 3 3 3 4 5 6
.
.
.

hmmm....

??so then _is_ there maybe some sort of morita equivalence here, or something ??? .... ??maybe simply (??...) something about ... grade (g1,g2) being the direct sum of grade g1 of the first coordinate module and grade g2 of the second coordinate module ... ??seems pretty likely, i guess ... ???...

??even so, i'm still confused... still seems like... ???progression from dimensional theory to ag theory can't make up its mind (or my mind... or something...) as to whether it preserves geometric sums ... ???or something ??? ...

??maybe it _is_ my mind rather than its own? ... because it's making me think that i should have the same confusion in lots of other contexts as well ... ??or something ...

hmmm, lots of confusion here ... ???"progression from dimensional theory to ag theory" ... as left adjoint part of "doctrine interpretation" ... ??? but could also think of "moduli stack of theory" as stand-in for theory, in which case ... ??well, something about (2,1)-category of such moduli stacks as opposite of corresponding (2,1)-category of theories, so ... ???from this viewpoint what was left adjoint seems like right adjoint ???....

something about ... ???in general, 2 distinct ways of getting right adjoint g from left adjoint f... namely, by taking g = right adjoint of f (??in this context something about "underlying poorer environment" right adjoint to "free richer theory" ...), but then also by taking g = "f^op" ... ???or something ???? ...as above ... ?? ...

(??what _about_ how these relate in case of adjunction coming from morphism of locally presentable categories?? .... and so forth ... ... ????.... hmmm, not particularly "the same" ??? ... ??or something???)

(??something about idea "propositional doctrine" ... ??supposed to mean a doctrine whopse theories are "propositional", sort of ?? ... ??with syntactic and semantic (2,1)-categories being actually just 1-categories.... ??or something ???....)

??anyway... ??so are we saying something like that... the right adjoint (2,1)-functor taking the moduli stack of a dimensional theory to the moduli stack of the resulting ag theory has the extra property not ordinarily expected of a right adjoint that it preserves sums ?? ...???or something ???

(??as opposed to something about whether the right adjoint (2,1)-functor taking an ag environment to its poorer underlying dimensional environment in turn has a right adjont??? ....???also what _about_ something about ... "the (...) decategorified analog that doesn't seem to work" ... ???something about ... nevertheless, there's an example nearby (or something ... ???...) of a functor with both adjoints, namely ... the inclusion functor from _ab gp_ to _comm monoid_
... ???or something??? ....)

??then what _about_ whether it preserves more general (weak...) colimits, and whether it in turn has a right adjoint ??? .... and so forth ... ????....


???then also similar questions about "de-toricization" ?? ...

??any analogy between "basepoint" of toric variety and "degenerate models" of dimensional theories ??? ...???and so forth ???? ....