Jason (jcreed) wrote,

It's always been a little mysterious to me why, from a category-theoretic perspective, the tensor product in linear logic is focally positive. I expect positive things to be colimity/left-adjointy things and negative things to be limity/right-adjointy things, and a monoidal functor is just something you plop down into your set of requirements, not something that evidently arises as a (co)limit or adjoint.

But stephen lack's "A 2-categories companion", page 4, has the following interesting passage: (emphasis mine)

Suppose V and W are monoidal categories and F : V → W is a left adjoint
which does preserve the monoidal structure up to coherent isomorphism.
There is no reason why the right adjoint U should do so, but there will be
induced comparison maps UA ⊗ UB → U(A ⊗ B) making U a monoidal
functor. (Think of the tensor product as a type of colimit, so the left adjoint
preserves it, but the right adjoint doesn’t necessarily.) In fact the monoidal
functor U : Ab → Set arises in this way
Tags: categories, math
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