?? real topologies of 2d affine toric varieties ... ???

??? bruhat cells for sl(2) .... generic as giving plane, coincide as giving cone ..... ???? ...... ??? real vs complex.... ??? z/2 vs z ... ??? .... ??? langlands / tate / fourier duality ???? ...... ??? stern-brocot ... ???? .....

??? ... map from cones of n-dim fan to congruences of (Z/2)^n ... ??? or in complex case, of (R/1)^n ... ???? some sort of nice functor from fan cone poset to "toruses" ..... ????? ..... ???? relationship to "cocone category" ?????? ..... ????.....

?? niceness of functor (...) here .... ??? ??lattice homomorphism ?? .... "level-preserving" ??? .... ???? ....

??? "geometric realization" .... ??? "kan extension" .... ???? .... ??? maybe sort of straightforward .... ??? .....

???? "weighted colimit" .... ???? ... ??? homotopy colimit ??? .... ????