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Stream: theory: category theory

Topic: 2-category of nat. transformations between sym. mon. cat.

Jean-Baptiste Vienney (Oct 14 2022 at 08:08):

We know that there is a 2-category whose:

• objects are categories,
• morphisms are functors,
• 2-morphisms are natural transformations.

Now, if $\mathcal{C},\mathcal{D}$ are two symmetric monoidal categories (in fact, it works with any category $\mathcal{C}$), I know that the category $[\mathcal{C},\mathcal{D}]$ of functors from $\mathcal{C}$ to $\mathcal{D}$ and natural transformations between them is a symmetric monoidal category whose:

• tensor product is given by $(\alpha \otimes \beta)_{A} = \alpha_{A} \otimes \beta_{A}$,
• unit is the constant functor equal to $\mathcal{I}$, the unit of $\mathcal{D}$,
• symmetry is given by $\gamma_{F,G}(A) = \gamma_{F(A),G(A)}:F(A) \otimes G(A) \rightarrow G(A) \otimes F(A)$ where $\gamma$ is the symmetry of $\mathcal{D}$.

It follows that the 2-category whose:

• objects are symmetric monoidal categories,
• morphisms are functors,
• 2-morphisms are natural transformations,

is more than a mere 2-category because if I see it as a category $\bf{SymCAT}$ enriched over the monoidal category $\bf{CAT}$, then for every symmetric monoidal categories $\mathcal{C},\mathcal{D}$, $\bf{SymCAT}[\mathcal{C},\mathcal{D}]$ is not only a category but a symmetric monoidal category.

What is the best name for this structure?

Kobe Wullaert (Oct 14 2022 at 08:44):

It seems to me that your are stating that SymCat is enriched over itself.

Jean-Baptiste Vienney (Oct 14 2022 at 08:52):

Oh yes thank you, that's exactly that! So, I have two questions:

• I'm wondering if maybe we have more coherences than is given by this?
• Is it written somewhere?

Kobe Wullaert (Oct 14 2022 at 08:58):

volume 2 of Borceux's handbook of categorical algebra probably has this result. Probably more general, in the sense that any (symmetric, closed) monoidal category is enriched over itself.

Jean-Baptiste Vienney (Oct 14 2022 at 09:23):

Thank you very much. But I'm not sure if it is exactly that because what I state is that the cartesian monoidal category of symmetric monoidal categories and functors is enriched over itself so there are two different tensor products in the story. Is it exactly a particular case of what you say? I don't know if I understand it correctly.

Kobe Wullaert (Oct 14 2022 at 09:52):

I didn't notice the cartesian part. Then the result I pointed to is indeed not relevant. So you consider SymCAT as full sub 2-category of CAT and you have concluded that that SymCAT inherits the symmetric monoidal structure of CAT? Is this what you mean?

Jean-Baptiste Vienney (Oct 14 2022 at 10:34):

I clarified my mind during the discussion. Yes, this is exactly what I mean.

Jean-Baptiste Vienney (Oct 14 2022 at 10:55):

I think that what I mean very precisely is that SymCAT as a full subcategory of CAT inherits the cartesian monoidal structure of CAT and is enriched over (SymCAT as a cartesian monoidal category).

Jean-Baptiste Vienney (Oct 14 2022 at 11:00):

(and not only over CAT as a cartesian monoidal category like a mere 2-category)

Mike Shulman (Oct 14 2022 at 16:14):

You don't want to conisder SymMonCat as a full subcategory of Cat; you want to look at only the (strong) symmetric monoidal functors. In this case you can indeed show that it is a closed symmetric monoidal 2-category, in an appropriate sense, and therefore enriched over itself. The internal-homs are the category of symmetric monoidal functors, with the induced symmetric monoidal structure you describe; the version you indicated that ignores any monoidal structure on the domain is instead the "power" or "cotensor" of SymMonCat over Cat.

Mike Shulman (Oct 14 2022 at 16:19):

Abstractly, this works because the 2-monad on Cat whose algebras are symmetric monoidal categories is "pseudo-commutative". This was studied by Hyland and Power in Pseudo-commutative monads and pseudo-closed 2-categories, and later the proof was simplified using skew structures by Bourke in Skew structures in 2-category theory and homotopy theory.

Mike Shulman (Oct 14 2022 at 16:21):

Neither the internal-hom nor the tensor product of SymMonCat coincides with the underlying internal-hom or cartesian monoidal structure of Cat.

Jean-Baptiste Vienney (Oct 14 2022 at 19:21):

Thank you very much for your explanations and references! That makes a lot for me to learn to have a better grasp on this subject. I'll try to understand all of this. You have clearly solved my problem.

Chad Nester (Oct 15 2022 at 09:07):

I found this to be very approachable.