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The term “linear category” currently redirects to “algebroid”, but Benton uses the term for a very different idea. For Benton, a linear category is
a symmetric monoidal closed category (L,⊗,I,⊸) and
a symmetric monoidal comonad (!,ε,δ,q) on L equipped with
monoidal natural transformations e,d with components eA:!A→I and dA:!A→!A⊗!A
such that
each (!A,eA,dA) is a commutative comonoid,
eA and dA are coalgebra maps, and
all coalgebra maps between free coalgebras preserve the comonoid structure.
He showed that this idea is equivalent to a symmetric monoidal adjunction between a symmetric monoidal closed category and a cartesian closed category.
Is Benton the only one who uses the term that way?
I expect he isn’t the only one, but I’d have to delve into the literature to be sure. I was glancing at Benton’s work not long ago.
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