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• CommentRowNumber1.
• CommentAuthorsanath
• CommentTimeOct 2nd 2014

Have added the “definition” of a symmetric monoidal $(\infty,n)$-category to the entry.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 2nd 2014

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.

• CommentRowNumber3.
• CommentAuthorsanath
• CommentTimeOct 4th 2014

I will try to do so this weekend. Is it OK if I first post it here, for review, and then edit the entry?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 4th 2014
Sure. And I am not trying to bug you, it should be an easy exercise. Essentially the text by Jacob Lurie which you copied contains it all, but, being a forum commnent, it is not written in the style in which it would appear in a textbook, with all the formalities spelled out.
• CommentRowNumber5.
• CommentAuthorsanath
• CommentTimeOct 5th 2014
• (edited Oct 5th 2014)

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 $k$.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 9th 2014
• (edited Oct 9th 2014)

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?

• CommentRowNumber7.
• CommentAuthorsanath
• CommentTimeOct 18th 2014

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 18th 2014
• (edited Oct 18th 2014)

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.)