?? relationship between tag-geometric and topos-geometric morphism .... ????? 2 contrasting ways of turning comm monoid hom into bi-action ??? ..... ?? and accompanying ways of trying to recover original hom from bi-action .... (??? issue of kinds / degrees of forgetfulness of turning-into processes here .... ??? ...... ?? also somewhat generalized here .... ?? = there ??? .... (?? no ?? ...) .... stackier situations .... ?? ... ?? ... ?? monoidalness structure vs cartesian monoidalness property ... ??? .... ?? ....) .... ?? and for one of the two ways, the turning-into process can be extended to include "localizations" .... ???? .....

?? so what are the alleged two ways ???

1 ?? [f(x),y] .... ???? .....

2 ?? ..... ??? ....

?? different bi-action, vs different way of getting bi-action to act on action .... ?? .... vs ... ???? ....

??? taking half-speed dynamical system of a dynamical system, vs taking double-speed ..... ?? ... ?? double-speed as tensoring with certain bi-action, but is half-speed also ??? ..... ??? maybe converse bi-action ??? ...... ????? .....

(?? funny the way we do seem to have managed to get those two confused ... ??? .... maybe "resolution" instead of "speed" here ??? .... ???? ....)

?? double-speed composed with half-speed .... ???? .... both ways ... ???? .....

?? hmmmm .... original-flavor hecke operator stuff going on here ??? ...... ?????? ...... ????? ......

??? vague heisenberg/hecke/murphy commutation flavor here ... ???? ....

?? hmmmm ... but what about monoid vs group here ??? .... ??? is there already something interesting going on in group case here ????? ...... ?????? ......

?? mono/polynomial vs laurent mono/polynomial here .... ??? .......