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1. • CommentRowNumber1.
   • CommentAuthormetaweta
   • CommentTimeDec 24th 2021
   • (edited Dec 24th 2021)
   • PermaLink
   Author: metaweta
   Format: MarkdownItexThe term "linear category" currently redirects to "algebroid", but [Benton](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.5454) 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 27th 2022
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   Author: Todd_Trimble
   Format: MarkdownItexI 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.

   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.