i want to take a stab here at describing the filteredly cocomplete category associated to a fan ...

...trying to use the "affine" case as a guide ...


the objects of the category should be the cones of the fan ...

given cones c1 and c2, a morphism from c1 to c2 should be a dual vector v such that for any x in the dual cone c1*, x+v is in the dual cone c2* .... ????...

???so for projective line, this seems to be giving hom table .... ????

N00
ZZZ
00N

???? or something ???

??really just saying that the objects are the dual cones and the morphisms the "translation maps" between them?? ... ??does closure (of fan ...) under intersection and "taking faces" lead nicely to filtered cocompleteness of the category of dual cones with "translation maps" ?