?? applying (?? ...) law "x^m = x^n" to toric geometry .... ???? for example m=3, n=1 .... ??? real spectrum .... ???? localization, co-/face, accidental topos .... ???? ....

?? "x^a * (x^b -1) = 0" ....

??? a = 1 or 2 ... b = 1 ... or 2 .... ????? ....

?????? loss of interesting z-gradings ???????? ...... ??? z-graded comm monoid st any ungraded localization satisfies x^m=x^n ... ???? ..... ??? ..... ???hmmmm, but what about z/p-gradings ??????? .... ????? .....

??? loops and tails .... constrained in certain ways .... ???? .... ??? laws for monoids vs for their actions ... ??? .... ???? .....

???? localization of semilattice ??? .... ??? is localization really flat (?? ...) in this generality??? how _does_ proof go?? ??? inverting _monic_ ?? ... ???and / or epic ???? inverting dominance relation in poset ... ???? .... ???? ..... ??? some monoid/action confusion here ?????? .... ???? filtered colimits .... ?????? ........ ??? "torsor" of [walking idempotent] monoid .... ?? colimit of infinite sequence of non-trivial endomorphisms of cayley action .... ?????

??? hmmm, but really just splitting idempotent here ??????? ..... ????? ....
???? some semi-recent discussion with todd ..... ???? splitting idempotents in semi-lattice-as-monoid ....??? .... ???? _was_ that in sort-of this context ???? ..... ???? the two (???? ....) cocone categories ... ???? .... ??? .....

??? TAG structure for actions of semilattice ... ??? .... ?? shortage of gradings by ab gp ... ??? shortage of non-affine possibilities from glueing .... ?????

???? etale groupoid .... ???????? ......

??? endo-ness as involved in relating "localization" (that is, inversion ...) to filtered colimits ???? .... ???? .....

???ab gp objects in (?? generalized ?? ...) accidental toposes here .... ??? ....

??? segal-morse category ... ????? .... ???? arnold / bott /appolonius .... ????

???toric geometry "based over z/n" .... ??? ..... ?? "galois-theoretic" aspect ???? .... ??? spectrum over commutative monoid under Z/n ; for example n=2 and (R,*) ... ???? ....

?? filtered colimit as generalization of retract .... ???? morita .... cauchy .... stable ..... ??????? .....