?? tensoring cocone with truth-value comm monoid t ... ?? then homming it into t also, and comparing that to combinatorial face structure of cone ... ???

?? ["subdivision morphism" between toric varieties] approached this way ... ???...

??? "dual to cocone face is cone face where the inequality it represents is the pullback of the fundamental inequality" ??? .....

"fundamental inequality" 0 <= p ???? ....

?? ... "fenchel ... " ??? ....

?? discussion with todd ... me at first thinking that cone / cocone duality was just about "N as schizophrenic between comm monoid and comm monoid" (???? ....) ... ?? but then todd pointing out what (?? ...) was wrong with that ... so then i suggested "0 <= f(x) galois connection between x and f" idea .... ?? possibility of distinction between these ideas as relating to some sort of "cohomology for comm monoids" ?? ... ????? ......

?? "folded schizophrenia" ... ??? torn between two identical copies of self ... ??? ....

??? but now ... "dual to cocone face is cone face where the inequality it represents is the pullback of the fundamental inequality" vs ... ?? contrasting idea ?? ... ... truth-values ... ????? ..... ??? or maybe not contrasting ?? ?? but ... i want to say / understand here about ... ?? how truth-values are fitting in here ????? .... ?? "zero on cocone face and "tribal infinity" off it" ?? ...

?? so what _was_ the anomaly (?? ...) that todd pointed out ??? ... ?? corresponding cone as not always precisely hom[cocone,N] ?? .... ?? example ?? ...

??? "boundary" fan of cone ??? .... ???? ..... ?? "child's drawing" and "through the looking-glass" ??? ..... "polytope" vs "combinatorial manifold" ... ?? ... "morse theory" ... ??? "removing stacky point to obtain projective space" ... ???? ...

?? through the looking-glass ... adjoining cones in fan, glued together along mutual face ... ???? ....

?? anti-fundamental inequality .... ????? ..... subdivision morphism ... ??? ....

?? case where adjoining cones have same "affine hull" ??? .... vs other case ... ??? .....

?? adjoining highest-dimension cones ... ??? ....