?? "x-ideal" ... "x-real" (?? "x-actual" ??? ...) ... ?? x "ramification
degree" ... ?? elliptic vs modular ... ??? ..... jugendtraum .... ???? ..... ?? "sub-x-ideal" .... ???? ..... ???? .....

?? see paper discussion from this morning ... ?? contravariance of galois correspondence as hint about "fourier duality" ??? .... ?? algebraic geometry ... ?? .... toric .... ??? ..... ???? .....

?? bit about mere subgroup / subfield correspondence vs more detailed thing .... ??? .....

?? "theory of relation from 1 to 1 which locally is either graph of map one way or the other" ... ???? .... ?? relation vs co-relation ?? .... ?? .... ?? ag vs g doctrine ... ??? ....

?? toric categories .... in some sense ... their universal properties ... in various senses .... ??? try to straighten soem of this out .... ???? spectrum of exotic toric stack over arbitrary comm ring ... ?? "richer doctrine" .... ????? ..... ag morphism with right adjoint strong monoidal wrt toric convolution .... ??? various adjunctions here ... comonoids .... ???? .....