?? specification of quotient torus, vs of "attachment information" ... ??? for example, 2d fan with single ray .... ?? the line of the ray, vs its direction ... ??? hmmm, should we then just reduce higher co-dimension attachment to co-dimension 1, on the grounds that the higher case is mediated by co-dimension 1 steps ... ??? vs ... ??? thinking about, for example, ways of attaching 1d torus to 3d ... ??? 3d fan formed from single "angle" ... ???

?? attachment ... ??? artin-wraith... and/or other stuff ... ???? ...

?? if intersection of cones isn't a mutual face then does that give "non-separation" ??? ... ??? ...

?? "newton polygon" .... "tropical geometry" ... relationship to toric geometry ... ??? .....

?? "supporting hyperplane ... " .... ??? ....



??? ?? "toric variety as built out of toruses" ... ??? then "non-linear fan" .... ??? ....

??? nerve of cover ... ???? simplicial complex as element of free distributive lattice ..... ????? free boolean algbera and (P,+)-spectrum vs (P,*) (??? ...) of toric variety ... ????? ....... ??? ....

?? "blowing up corner of polytope to extra face" ?? .... ??? ... poincare dual ... ??? ....