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Stream: theory: category theory

Topic: linear natural transformation

Nicolas Blanco (Oct 02 2022 at 10:27):

Warning: I am using $\oplus$ instead of the usual par for the multiplicative disjunction because it seems that Zulip does not recognize \parr
In their paper "Linearly distributive functors", Cockett and Seely define a 2-category of linearly distributive categories, linear functors and linear natural transformations.
A linear functor consists of a pair of a lax $\otimes$-monoidal functor $F_\otimes \colon \mathcal{C} \to \mathcal{D}$ and a colax $\oplus$-monoidal functor $F_\oplus\colon \mathcal{C} \to \mathcal{D}$ with some strength natural transformations relating $F_\otimes$ and $\oplus$ and $F_\oplus$ and $\otimes$ subject to some coherence conditions.
A Frobenius functor is a linear functor where $F_\otimes = F_\oplus$ and the strengths are trivial.
Any monoidal functor $F$ between $\ast$-autonomous categories induces a linear functor with $F_\otimes := F$ and $F_\oplus := F((-)^\ast)^\ast$.
Furthermore, there is an equivalence between the category of linearly distributive categories and Frobenius functors and the category of two-tensor polycategories and functors.

However, when we try to define linear natural transformations between linear functors it gets more complicated.
In the paper, the authors define a linear natural transformation as a pair of monoidal transformations $\alpha_\otimes \colon F_\otimes \to G_\otimes$ and $\alpha_\oplus \colon G_\oplus \to F_\oplus$ (notice the reversed direction) with some extra coherence laws.
Then, given two monoidal functors between $\ast$-autonomous categories, a monoidal transformation $\alpha \colon F \Rightarrow G$ induces a linear transformation $\alpha_\otimes := \alpha$ and $(\alpha_\oplus)_A := \alpha_{A^\ast}^\ast$.
However, as far as I can see the equivalence between the category of linearly distributive categories and Frobenius functors and the category of two-tensor polycategories and functors does not extend to a 2-equivalence with this notion of linear natural transformation. Instead, one would need to define linear transformations as pairs of monoidal transformations $\alpha_\otimes \colon F_\otimes \to G_\otimes$ and $\alpha_\oplus \colon F_\oplus \to G_\oplus$. This is the choice made on the nlab page
Is anyone aware of other references where linear natural transformations are used and what is the definition they take? Also is the 2-equivalence between ldcs and two-tensor polycategories discussed somewhere? More generally do you have a strong feeling towards one of these definitions?

Mike Shulman (Oct 02 2022 at 17:57):

My strong feeling is that the polycategorical notions are the correct ones.

Mike Shulman (Oct 02 2022 at 17:58):

By the way, for par you can just paste in the unicode: ⅋

Nicolas Blanco (Oct 03 2022 at 11:38):

Thanks. I have a similar feeling. But since the only reference that I know of about this is the paper by Cockett and Seely, I was wondering if I was being too biased towards polycategories.

Matteo Capucci (he/him) (Oct 03 2022 at 17:57):

Maybe it's completely wrong but have you looked at Grandis and Parè Intercategories: a framework for three-dimensional category theory? One of their chief examples is duoidal categories, which are 'lax' distributive categories. Duoidal categories are one-object intercategories, so they span a full sub-intercategory of the intercategory of intercategories (!). This intercategory has lax-lax, colax-colax and colax-lax as 1-cells and suitable notions of transformations as 2-cells, with a commutativity condition for 3-cells IIRC.

Mike Shulman (Oct 03 2022 at 18:11):

Duoidal categories combine a pair of monoidal structures in a very different way from linearly distributive ones.

Matteo Capucci (he/him) (Oct 03 2022 at 18:17):

I see

Nicolas Blanco (Oct 04 2022 at 13:17):

Thanks @Matteo Capucci (he/him) Although duoidal categories combine monoidal structures in a different way, this paper still seems interesting. Another one on my really long to-read list!