i want to think outloud here about the "tensor product" functor on the co-cones of a co-fan ...

one reason that this functor should exist is as the manifestation at the model level of the alleged essential geometric morphism from t^2 to t for which the tensor product of toric quasicoherent sheaves (preceded by the universal bicocontinuous functor) is the extra left adjoint, where t is the accidental topos of the corresponding toric variety. that's a bit circular, though, since my reason for guessing that such an essential geometric morphism exists is mainly because of its manifestation at the model level.

anyway, i have a couple of further guesses about how to think about the alleged tensor product of co-cones:

1 as a sort of categorification of the intersection operation on the poset of cones of the fan ...

(???confusion/problems here ??? .... ??? unit element for intersection as "total subset" ... ???? whereas that doesn't seem to qualify as a cone, even in affine case ???? .... ???? .... ????hmmmmmm .... ???maybe slightly tricky ???? .... when there's a largest cone (that is, the affine case ... ??? ...), it can play (at ?? ...) the role of "total subset" ... ???sa... ?? this as maybe sort of nice, in suggesting something fishy about unit element for intersection of cones even in those case where it does exist ... ???sa ... ???as not preserved by inclusion into semi-lattice of all (??? ... ???even those including non-trivial antipodal pairs ??? ... ???) cones ... ????)

(hmm, so what about co-cone viewpoint on above parenthetical discussion?? ... span of co-cones ... ??? binary such as "safe", nullary such as ... ???? ..... ???? .... ...hmmmm... co-slice ... ?? ... ??vaguely reminds me of bit about "every slice cat is a topos / distributive lattice" ... asf os...)

2 as "tensor product" of categories enriched over the whole lattice ... ??? ...