?? so let's take a stab at inventing "generalized kan extension" here ... ?? ...

?? hmm, but what do we mean by this, really ... ??? ... ?? maybe actually not a good name at all for what i think i have in mind ?? ....

?? "generalized day convolution" is (?? so far ?? ...) just supposed to generalize the idea of day convolution wrt an actual monoidal structure, right ?? ... ?? or maybe just semi-monoidal, i guess ... ???? .... ?? so maybe already "generalized day convolution" was a bad name ??? ...

?? "kan extension" as adjoint to "restriction functor" ?? ...

f : x -> y

g : x -> z

?? "kan-extend g along f" as adjoint to "restriction" functor from [y,z] to [x,z] given by "pre-composing by f" ... ??? ...

?? maybe x and y should be small ?? ....

?? specialize to z = _set_ ... ?? ... ?? then left kan extension as extra left adjoint of essential geometric morphism between presheaf toposes ....

?? whereas i'm interested in extra left adjoints of essential geometric morphisms between more general grothendieck toposes ?? ... ?? ...

?? hmm ... but i also have this vague feeling that there's some big relationship between "kan extension" and "tensor product" that i'm forgetting / not quite seeing here .... ??? ... ??? well, what about "tensor product as day convolution" ???? ..... ??? ..... ?? hmmm, co-/ends and mac lane's book ... relationship to kan extension ... ?? "all concepts are ..." ... ??? .....

_n-torsor_ X _n-torsor_ -> _n-torsor_

?? "tensor product goes the same way as the models go, but ... " ??? .... ??? ....

??knowing how to tensor the "strict" n-torsors, and knowing how to express the non-strict ones as filtered colimits of the strict ones ... ??? ... ?? should allow us to describe nicely concretely how to tensor the non-strict ones ... ??? ...

?? seems like ... ??? in the tensor product of non-strict torsors x and y, the stuff tha gets inverted (?? does this really parse nicely here ?? ...) is precisely what results from the stuff that you inverted to get x and the stuff that you inverted to get y .... ???? which does sound morally something like "intersection of basic toric zariski opens" ... ???? ..... ?? relationship to tensor product of quasicoherent sheaves associated to those opens .... ????? this confusion again ??? ..... ?? "failure of generalized yoneda embedding" ??? .....