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I know the question of what is a symmetric monoidal category has shown up here.

I was wondering if there was a more informal way of describing a symmetric monoidal $(\infty, n)$-category as a "sequence" of symmetric monoidal categories.

Or perhaps is there a notion of a strict monoidal functor between strict monoidal $n$-categories? If so, can we somehow take this definition and "weaken" in to arrive at an informal definition?

Lastly, if either above are affirmative, can we then describe a symmetric monoidal functor between them in an informal way?

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  • $\begingroup$ What do you find unsatisfying with the definition? Right now the question is extremely vague. Would you be satisfied by a description of commutative monoids in a general (∞,1)-category (as, for example, the (∞,1)-category of (∞,n)-categories)? $\endgroup$
    – Denis Nardin
    Sep 13, 2016 at 22:09
  • $\begingroup$ @DenisNardin Well I only have at this moment a working definition of (\infty,m)-categories, namely as a "weakened" strict n-category as n->\infty with k-morphisms inveritible for k > m. Perhaps this is wrong. I was just wondering if there was a way with my current knowledge (not knowing homotopy inverse limits, E_{\infty)-operads, model structures, etc.) to have some intuition for what a symmetric monoidal (\infty,n) category is. $\endgroup$
    – SWV
    Sep 13, 2016 at 22:24
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    $\begingroup$ Informally, a symmetric monoidal $(\infty, n)$-category is an $(\infty, n)$-category equipped with a monoidal structure which is commutative in a suitably homotopical sense (in particular, in a way which gives it extra structure rather than just an extra property). Is this enough or do you want more, and if so, what more do you want? $\endgroup$
    – Qiaochu Yuan
    Sep 13, 2016 at 22:37

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