?? "toric ag with exponentiation" ??? ....

?? "product-exponentials algebra" .... ??? as "comm monoid with hom to its own endomorphism monoid" ??? .... ??? is that all it is / should be ??? .... ?? categorifiying, and/or throwing addition into the mix ??? .... ??? quandle echos here ??? .... ?? lawvere / schanuel / joyce / freyd / yetter ...

?? "action of underlying comm monoid of pe alg x, compatible in certain way with x's "exponentiation" action of u(x) on itself" .... ??? in certain way imitating way exhibited by u(x)'s cayley action ???? .....

?? x's exponentiation action of u(x) on itself as nicely / canonically extending (...) to action of u(x) on (comm monoid, action) pair (u(x), cayley(u(x)) ?? ....

?? by ... ??? thinking of "cayley" as lawvere theory morphism ?? ...

?? with left adjoint "cayleyfication" ?? ... ??? ... (?? idea for actual systematic terminology here ??? .... ???? ....) ....

?? concretely .... ???? .....

?? hmm, so are we suggesting here that there should be _another_ ("exponentiation" ??? ....) action of m on a required, besides the usual ("multiplication" ...) one ??? ....

(a1 * m1) ^ m2 = (a1 ^ m2) * (m1 ^ m2) ???? ....

?? "semi-direct product using inner action as actually direct" trick here (??? ....) ??? ... ???? ... ???? ....

?? .... schanuel .... ?? tarski .... ??? ....

?? semi-direct product as weak and / or maybe lax colimit ?? .... ?? presheaf topos over weak / lax colimit site category as ... ??? ....