?? lax interchange map from cartesian product outermost to convolution outermost corresponding to "adding solutions to x+y = k1 and x+y = k2 to get solution to x+y = k1+k2" ???? ..... ??? does this make any sense ?? ...

?? hmmm ... some fourier-duality confusion here ... pointwise product vs convolution ... which of these cartesian product qualifies as ... ??? ..... ?? maybe ok though ??? ....

?? cartesian product as "pullback along diagonal" ... ???? ..... ??? convolution as "pushforward along codiagonal" .... ???? ..... ???? ....

?? pointwise tensor product of graded vector spaces, vs convolution tensor product ... ???? .....

(a toric b) ordinary (c toric d) -> (a ordinary c) toric (b ordinary d) ... ??? ....

(a cartesian b) convolve (c cartesian d) -> (a convolve c) cartesian (b convolve d) ... ??? ....

hmmm .... ??? ..... ?? "components given by functoriality of convolution, applied to projections" ??? ....

(a convolve b) cartesian (c convolve d) -> (a cartesian c) convolve (b cartesian d) ... ?? ...

??????? .......


(.... , a_-1 * b_1 , a_0 * b_0 , a_1 * b_-1 , ...) ...... ???? .....


??? hmmm, so maybe arrow really is backwards from expected here, due to duality flip .... ???? .....

?? skyscraper sheaves here .... ??? and how they get along with duality ... ?? ....

?? this (?? ...) lax interchange property as vacuous on topos side, but not on ag side ??? ... ??? .... ?? adjointness here ???? .....

??? cartesian smc with extra tensor product ... ?? vs smc with extra tensor product and lax exchange map .... ???? ....

?? other arity lax exchange properties here ?? .....