???? .... accidental geometric morphism .... factorization ... distributivity .... stuff .... flag geometry .... extension .... ??????????????????????? ....

?? confusion about first vs second vs extremes against middle here ... ??? ....

?? trying to understand geometric morphisms _and 2-morphisms between them_ exploiting lowbrow or highbrow treatment of "factorization system" here ...

?? how 2-morphisms get along with factorization of 1-morphisms .... ??? possible funny interaction between variance choice at different levels ?? ...

?? hmmm .... canonical / natural / functorial / 2-functorial (... ?? ...) vs unique (??up to ... ??? ...) factorization ..... ???? .....

?? "strictly canonical factorization" ... ??? including .................... strict equality of intermedite object .... ??? for example canonical surjections and injections, or injections and canonical surjections ..... ???? ....

"borrowing/[acquiring by contamination] [canonical-but-not-unique]ness (... ??? perhaps somewhat different-from-usual (?? perhaps more literal ... ??? explicit act of canonization without which the choice wouldn't stick out ... ?? issue of morphisms being required to preserve the previously (?? ...) arbitrary choice ... ??? ...) meaning of "canonical" here, though might be able to shoehorn it in ... ???) from that of coequalizer of er and/or equalizer of co-er ..." ... ??? .... ?? actually that as funny way to try to put it because of uniqueness for er in turn / in the first place ... ??? ....

?? nonce braiding in monoidal category of endo-modules of discrete category vs of non-discrete .... ???? ...

?? anyway ... ??? (2- ??)category with all morphisms endo ... objects as toric varieties, morphisms as "toric" line bundles ... ?? use in "factorization system" ... ??? as "substrate" and / or as one of the composants ???? .... ?? vague resemblance to .... ?? certain semi-famous examples of group and/or monoid extensions / semi-direct products ... ?? automoprhisms acting on translations ... ??? ..... ??? substrate vs superstrate here ?????? ...... ???? ..... "jordan-holder" and nonce-braiding ..... ???? ..... ?? vaguely reminding me of question coming up in flag geometry ..... ??? "parabolic induction" ??? ..... "what is the extra stuff that [blank] has?" ... ?? having trouble remembering exactly how to formulate the question at the moment, though pretty sure that i could reconstruct it ... ??? "residual geometry" .... ?? in "residual geometry" situation there's some obvious "extra stuff", but then it becomes clear that that's not _all_ the extra stuff ... thus provoking that question .... ??? ....

??? anyway, could certainly try to take very general (?? "highbrow" ?? ...) approach here, but should really mostly sticking with lowbrow for now i think ...

(?? general ... ?? ...)

(?? "educational experiment" ... ?? ...)

?? first try to outline how 2-morphisms between geometric morphisms fit in with surjection/injection factorization in general topos context, then try specializing/adapting (?? ...) to case of accidental topos ... ??? ...

?? so somewhere pretty near beginning of paper (?? new section on geometric morphisms between accidental toposes ?? ...), maybe should introduce "toric" line bundle in bootstrappy way ... ??? as comonoidal wrt toric convolution ??? .... ??? hmmmm, maybe a bit _too_ bootstrappy (in the bad sense ...) ... ?? well, the idea of quasicoherent sheaf of actions instead of of modules is accessible enough, i guess .... ???? ... ?? hmm, "invertible sheaf" terminology as maybe helpful here ?? ...

left exact comonad on geometric domain topos ....

idepotent left exact monad on geometric codomain topos ... .... ?? ...

????? maybe something nice about ... idea of "first tensor by localization of unit object, then tensor by "(relatively- ... ??? ...)invertible" object" ... ?? i meant "nice" just in ... having two vaguely allied things ("localization of unit object" and "invertible object" .... ??? ...) somewhat lining up next to (?? ...) each other in "factorization system" ... ??? .... but ... ?? now also .... ??? ... ?? idea of ... ?? understanding maybe nice interaction here ... ??? this idea of "relative invertibility" .... ?? i was going to suggest "torsor" approach here (?? maybe guided to some extent by "cohomological" (?? ...) spirit ?? ...), but ... ??? maybe (?? categorified ?? ...) addition/multiplicaiton confusion here ??? ....

(?? so ... ?? _are_ "bundles of 1d affine spaces" interesting here (?? ...) ??? ..... ???? and how much did we ever really know about them ?? ... ?? certainly there was _something_ that we thought that we understood about ... short exact sequence approach to "affine vector space" ... and some sort of "cohmology" idea connected with it ... ??? ...)

(?? cockett .... ???? glueing ... ??? trying to get general ag case to work somewhat nicley ... ?? like tag case ??? .... ??? and trying to interpret this as secret behind ... ?? certain stuff ... ?? topos theory in ag ... ?? .... glueing in ag ... ??? lots of things here seem to be having trouble trying get to fit .... ???? ....)