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Have added the “definition” of a symmetric monoidal $(\infty,n)$-category to the entry.
Thanks.
Wanna give it a try spelling out the actual definition, in math-style?
“A symmetric monoidal $(\infty,n)$-category is …”
And then:
“By the general discussion at k-tuply monoidal (n,r)-category, one expects this definition to be equivalent to …”
Let me know if you feel like giving it a try. The idea is to get a feeling for writing actual precise definitions in $\infty$-category theory. If you feel unsure, let me know and I’ll give some more hints.
I will try to do so this weekend. Is it OK if I first post it here, for review, and then edit the entry?
Here’s my attempt:
Definition: A symmetric monoidal $(\infty,n)$-category is an $(\infty,n)$-category with an $\mathbb{E}_\infty$-action. Thus a symmetric monoidal $(\infty,n)$-category is an $(\infty,n)$-category that is a $k$-tuply monoidal $(\infty,n)$-category for any $k$.
Sorry, I had been out of action for a while.
You might want to think about the following, generally:
What precisely does it mean for anything to be equipped with an $E_\infty$-action?
What do you need to prescribe for that to be well-defined? Do you need the $(\infty,n+1)$-category of all $(\infty,n)$-categories? Do you need to specify any monoidal structure?
I apologize for the late response. If anything is equipped with an $E_\infty$-action, then we can assume that it has a commutative algebra structure. We need a bi-$(\infty,n)$-functor $\otimes:\calC\times\calC\to\calC$, that satisfies the axioms for monoidal categories, that is commutative. I do not think that we need the $(\infty,n+1)$-category of $(\infty,n)$-categories, but I think I might be wrong.
All right, so first, as you suggest, we just need the $(\infty,1)$-category of $(\infty,n)$-categories. This is because all the higher homotopies involved in passing from the concept of a commutative monoid to that of a $E_\infty$-object are all invertible higher maps and this translates into only needing invertible higher “transfors” between $(\infty,n)$-categories for describing an $E_\infty$-structure.
Second, as you implicitly indicate in your formula, we need the cartesian monoidal structure on the category of $(\infty,n)$-categories.
So to start stating the precise definition of symmetric monoidal $(\infty,n)$-categories as $E_\infty$-objects, we first of all need the concept of symmetric monoidal $(\infty,1)$-category. Then we need to specify the specific case
$(Cat_{(\infty,n)}, \times) \in SymmMon(\infty,1)CAT$given by the $(\infty,1)$-category of $(\infty,n)$-categories equipped with its cartesian monoidal structure.
The theory of symmetric monoidal (infinity,1)-categories is developed “from scratch” for instance in Higher Algebra. The example of $(Cat_{(\infty,n)}, \times)$ may be obtained for instance via the cartesian monoidal model category presentation of Theta n-spaces.
Once one has any symmetric monoidal $(\infty,1)$-category $(\mathcal{C}, \otimes)$, then one may ask about equipping its objects with the structure of an algebra of the $E_\infty$-operad. A traditional way of saying this, which makes clear the need to specify $(\mathcal{C},\otimes)$ beforehand, is that such an $E_\infty$-structure is a homomorphism of $\infty$-operads $E_\infty \longrightarrow End(\mathcal{C},\otimes)$ to the “endomorphism operad” of $(\mathcal{C},\otimes)$. What this means in the $\infty$-category context is the content of the first few chapters in Higher Algebra. The end result is a concept of commutative monoid in a symmetric monoidal (infinity,1)-category (I see that this entry does not explain anything yet, somebody needs to put in information here, too).
So there is a bit of technology involved in setting this up. Once one has this, then one may say:
A symmetric monoidal $(\infty,n)$-category is an $E_\infty$-algebra object in the symmetric monoidal $(\infty,1)$-category $(Cat_{(\infty,n)}, \times)$ of $(\infty,n)$-categories equipped with its cartesian symmetric monoidal structure.
(Notice that one such $E_\infty$-structure is not just an $(\infty,n)$-category $\mathcal{X}$ equipped with an $(\infty,n)$-functor $\mathcal{X}\times\mathcal{X} \longrightarrow \mathcal{X}$, but is in addition an infinite tower of higher transfors, those exhibiting the commutativity of this product up to higher coherent natural equivalences.)
That would be one way to start with a precise definition.
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