?? "fan spectrum" ( = (P,+) and/or (N,+) spectrum ...) vs real spectrum of toric variety ... ??? inclusion (?? ...) of former in latter ... ?? some confusion concerning ... ?? "poincare duality" schematic (?? ...) geometric interpretation of fan .... difference of toric zariski opens .... ?? may be single point in real spectrum but more in fan spectrum ... ??? because there's fewer points being _subtracted off_ in fan spectrum ???? ..... ????? ......

?? "zariski topology" here ?? .... ??? .... ??? "control" ??? .... ??? .... ...... ???? .....

??? detailed explicit look at how fan spectrum fits in real spectrum ... ???? .... ?? maybe especially in subdivision / blow-up case ... ??? .....

?? (P,+) as 1/4 of (R,*) ????? ..... ????? ...... ?? sort of ??? .... ?? but .... ?? [P^1]_(P,+) as about 1/2 of RP^1 ... ??? ..... ]0,1[, 1, ]1,infinity[ ... ????


?? cocone category of toric variety as (??maybe op- ??? ...????)lax colimit ... ???? how this relates to .... lax limit status of category of action-sheafs = category of set-valued functors on cocone category ... ??? .... ???? "borderline eckmann-hilton" .... ???? "pro-eckmann-hilton" ???? ....... day convolution .... ???? ......