??ok, so what _about_ the models of the accidental topos of the projective line??

??also... ??what about model(t)^op -> t ... given by ... ???or something ??


for example consider t := the object classifier ...

a model here is just a set m ...

which (??contravariantly??) gives a functor from _finset_ to _set_, namely "x |-> x^m" ... ??...

??have i been getting terminology "cone" (in fan of toric variety) a bit mixed up ??? ... ??or something ?? ...

???hmmm, so what about something about ... ??filteredly-cocomplete category x ... ???yoneda embedding x^op -> [x,_set_] ... ??but then composed with functor [x,_set_] -> [x,_set_]_filtered-colimit-preserving which is right adjoint part of "surjective geometric morphism" ... something about "cofree coalgebra of comonad", or something ??... ???_is_ this the way it goes ???....

??something about ... ???comonad on _set_^2 ... (a,b) |-> (aXb,aXb) ... ???and so forth ???... ???something about having left adjoint monad (c+d,c+d) <- (c,d) ??? or something ???.... ????some level slip about "idempotence" here ??? ???"idempotence" of factorization system (and so forth ...) vs ... ???idempotence of monad or comonad associated with one of the adjunctions that an adjunction is factored into ?? ... and so forth ... ??... ???so what _about_ the "model objects" in a topos ??? ... that is, simply the image of the embedding (??or something...??) model(t)^op -> t ... ???...