?? "left-exact tensor product of quasicoherent sheaves" .... ??but no unit-object ???.... ??except in affine case ??? .... ???? .....

toric case ....

x ab gp ....

_set_^[x^op] -> _set_^[[xXx]^op] =?= _set_^[x^op] # _set_^[x^op] .... ???...

???what's going on here ???....

??no "co-day convolution" ??? ..... ?????......

???or maybe not "left-exact" ???? ..... ????? ....

?? m monoid .... ??? _set_/m -> _set_/[mXm] .... ????......

???"day convolution of sets over m, thought of as pre-sheaves over discrete category m" .....?????....

_set_/[mXm] =?= _set_/m # _set_/m ... ????...

_set_/[mXm] <- _set_/m X _set_/m ..... ???"universal bi-[2-linear]" ????... ??"2-linear" here as "cocontinuous" ??? .... ???? .... ???hopf object in (_set_,X) going to hopf object in (_topos_,#) ??? .... ???.... ???via s |-> _set_/s ??? .... ????.... ???? ...... ??? ......

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

g = z/2 ....

g as discrete monoidal category ....

addition in z/2 as functor z/2 X z/2 -> z/2 between discrete categories ....

_set_/(z/2) -> _set_/(z/2 X z/2) .... pair (f0,f1) of sets goes to "square" of sets (f00=f0,f10=f1,f01=f1,f11=f0) ... ????....

???right adjoint to this???

(f0 = f00 X f11, f1 = f10 X f01) ????.......

_set_/(z/2 X z/2) <- _set_/(z/2) X _set_/(z/2) ???? ???"universal bi-[cocontinuous]" ((h0,h1),(j0,j1)) |-> (f00 = h0 X j0, f10 = h1 X j0, f01 = h0 X j1, f11 = h1 X j1)

??then compose??

((h0,h1),(j0,j1)) |-> (h0 X j0 X h1 X j1, h1 X j0 X h0 X j1) .... ?????......

(h0 X j0 X h1 X j1 X k0 X h1 X j0 X h0 X j1 X k1, h1 X j0 X h0 X j1 X k0 X h0 X j0 X h1 X j1 X k1) .... ?????