i wrote a mathematica program to draw toric varieties in a certain way...

this exercise suggests various ideas to me, but i'm having a bit of trouble articulating them ...

presheaves on ... ??certain subcategories of _comm monoid_^op .... ????.... ???"geometric realization" process for these ... given by co-presheaf .... ???toric variety as built out of some sort of "cubes" ... ???some sort of "oblique" (????...) morphisms ... ??? "face" ... ??? .... ??? "degeneracy" ... ???? .... ??? ... "localization" ... "local isomorphism" ... ??? ...

???open cubes vs ... ???half-open ... ??? .... ???? ..... ?? ???grothendieck topology here ??? .... ???? ...... ???....

... ???non-toric analog ... ???geometric realization of preseheaf on _affine scheme_ as space (???....) of some kind ... ??? ... given by co-presheaf ... ???examples??? ...

presheaf _comm ring_ -> _set_

co-presheaf / "model" _comm ring_^op -> _set_ .... ???"galois shapeshifter" ??? .... ??? ....

contravariant hom functor of given comm ring ... ????maybe with "topology" ???? ... ??? ......

??not quite sure how right-track this "cubic" idea is yet .... ???? .... need to think more about toric glueing ... ??? ....