?? so let's try understanding all of the filteredly cocontinuous functors from for example the model category of _set_^z to the model category of _set_^x for some (nice ... toric-geometry-wise ... ?? ...) commutative monoid x, for example ... ?? ... treating exponents here as single-object categories ... ?? ...

this should be in direction that includes ... ?? just plain functors from z to x ..... ??? ..... with toric geometric interpretation as some sort of nice "toric map"s from spec(x) to spec(z) .... ?? so ... ??? we're somewhat expecting / hoping that the general geometric morphisms here are something like "nice partially defined toric maps from spec(x) to spec(z)" .... ???? ....

??? divisor of a "toric meromorphic function" ????? ...... ????? ....

?? " ... fibration ... " ???? .... ???? equivalence between slice categories ... ??? ..... ?? relationship to stuff that cockett may have been hinting at ?? ....

?? so given morphism m from z to groupization of x ... ??? .... ??? consider morphism from z to .... "least groupized version of x for which m exists" .... ??? .... ?? hmmm, but then there's also morphisms from z to "unnecessarily groupized versions of x", is that right ??? ... ?? do these really give distinct geometric morphisms ?? .... ???? .....

?? presumably issue of "do you get to explicitly specify the domain of definition of the partial map ?" .... ????? .....

??? so for example x := n .... ?? zero hom from z to groupization(x) here ?? ... ?? as only example where "unnecessary groupization" is available ??? .... well, in fact only example of hom z -> n, which maybe is more or less saying the same thing ??? ...

??? so .... ?? given n-set s ... ??? construed as filteredly cocontinuous set-valued functor on model category of _set_^n ... ?? .... ??? pull back along two allegedly somewhat different allegedly filteredly cocontinuous functors from model category of _set_^z to model category of _set_^n .... ??? .... ?? to get a z-set, more or less ??? ...

?? "assign to a z-torsor t the set s, and to a z-torsor iso the identity map of s" .... ???? .... ??? vs "assign to a z-torsor t the set "s tensored over n with z", and to a z-torsor iso the identity map of "s tensored over n with z" ..." ... ????? .... ?? really does sound like two inequivalent functors (from ... to ... ?? ...) ... ??? are they really both lex left adjoints ?? ....

?? "assign to an n-set s the z-set "s with trivial z-action"", vs "assign to an n-set s the z-set "[s tensored over n with z] with trivial z-action" ... ???? .... ?? bi-action interpretations here ??? ...