?? was trying to write up "toric proj" stuff "from filteredly cocomplete picture" when noticed complications .... ??? ....

?? relationship to accidental geometric morphisms associated with toric line bundles ... ?? ...

?? formula / model confusion here associated with "mayhem" ?? ... ??? ....

?? but even more ... ??? bit about "inverse image preserving toric affineness" ... ??? .... ??

?? slice topos approach here ??? ....

?? maybe loss of ... ?? being in nice simple case where "filteredly cocomplete picture" sees everything ... ??? ...

?? double negation topology on graded action topos ??? .... ???? .....

?? so ... ?? slice topos of _set_^[_n_^2] .... ??? over _z_ ??? ....

?? so ... can we make some sort of table of what we think the hom-cats are like .... and the composition functors ... ??? ... ?? hmmm, maybe lots of ... identification of model objects taking place ... under various "localizations" .... ?? try straightening it out ... ??? ...

?? hmm, so maybe ... ?? ... the models of the graded action topos are .... well, "z+1+1+1" ... if you know what i mean ... ??? ....