Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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?
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.
1 to 2 of 2