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• CommentRowNumber1.
• CommentAuthormetaweta
• CommentTimeDec 23rd 2021
• (edited Dec 23rd 2021)

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 $(L, \otimes, I, \multimap)$ and

• a symmetric monoidal comonad $(!, \epsilon, \delta, q)$ on $L$ equipped with

• monoidal natural transformations $e, d$ with components $e_A:!A \to I$ and $d_A: !A \to !A \otimes !A$

such that

• each $(!A, e_A, d_A)$ is a commutative comonoid,

• $e_A$ and $d_A$ 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?

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJan 26th 2022

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.