i want to take the theorem that you've proved about the universal property as an ag theory of the coherent sheaves over P^n and re-phrase it in a way that makes sense for the projective space of an arbitrary (or almost arbitrary...) vector space... on the theory that stating it in such a more invariant way may help to reveal what's really going on...
(eventually it may be possible to state the theorem so as to apply not merely to an external vector space but to any suitably nice (in particular dualizable?) object in an ag theory...)
so let V be a dualizable vector space...
then consider the ag theory of a line object L equipped with a morphism e into the "internalization" of V ... such that ... ???....
???so what about... ??symmetric algebra of dual of dualizable object... ??something about taking its graded modules... (???in "internal" sense???) ??and so forth, or something ... ???... ?????
coincidentally, martin asked me a question about something like this...
t ag theory...
x comm monoid in t ...
t -> _x-module_
y |-> y tensor x ...???
???"t-model ew homomorphism from x to the unit object" ...??? or something???....
x tensor x -> x ...
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