??so consider "the algebraic-geometric theory of an affine line's worth of invertible vector spaces, equipped with a nice way of tensoring them riding addition"?? ...

...and so forth ... ???....

...also the "multiplication" case ... ???....

??"invertible module over the affine line, symmetric monoidal wrt additive convolution" ... ???or something...

??to what extent _is_ all this nicely and unambiguously expressible in the doctrine (and/or in some related doctrines... ??...) ?? ...

??how does this relate to geometric interpretation of representations of borel subgroups??... and so forth... ??d-modules, and formal translation group vs actual translation group ... and so forth...

??how does this relate to "representations" (...) of some kind of commutative and/or cocommutative hopf algebras??? ... and so forth...

??how does this relate to "the regular representation picture of a g-torsor"?? ... and so forth ... ???...

...??something about "compactness" issues?? ... or something... ... "coherent..." ... ??....

??something about preservation of "[affine line]-indexed sums"?? ... and so forth...

??something about relationship between ["theory of an x-indexed family of..." for x the model stack of an algebraic-geometric theory] and [internal hom between algebraic-geometric theories... and so forth...]??... ??and... "algebraic stacks" ... ??something about geometric right-universal property of an algebraic stack... involving family indexed by another geometric stack... or something... ??...

??what about decategorified analogs here of.. ??situations where you sort of hope to have internal homs in some generality but they turn out to exist only in rather special cases... ??or something?? something about exponentiable affine schemes and so forth??... something about expressibility of ... ???....