so consider the symmetric monoidal finitely cocomplete algebroid t given as the opposite of the algebroid of fg free modules of the base field (or something...) ... ???what is this the algebraico-geometric theory of??? ... or something...

consider the algebraico-geometric morphism from the initial algebraico-geometric theory to t...

the morphisms invertibilized by this functor are... ??what??... ??is there something here about matrixes of determinant 1 ??? ???or something like that?? ??but what about non-square matrixes??? hmmm...

hmm, now i'm getting a bit confused about the relationship of this to stuff that we thought we almost understood about the algebraico-geometric theory corresponding to a "galois stack", or something like that... ??...

??something about normal vs non-normal extensions and property vs structure... or something ... ???... "axioms" ... ??...

??hmm, something about invertibilizing morphisms between free modules vs between not neessarily free ... ???or something??...
??hmm, what about something about "matrixes between matrixes" in the sense of, given a domain r1 X g1 matrix m1 and a co-domain r2 X g2 matrix m2, a g1 X g2 matrix "preserving the relators" ... i was going to suggest trying to generalize determinants to this context, but that doesn't seem quite right... ??something about "witness for preservation of each basic relator" as giving commutative square of matrixes??


let me remind myself that i think my original motivation for starting to think about this was to experiment with the idea of "extending distributivity by distributivity" or something like that... ??...