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11 votes
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I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-categ...

First, as Rune pointed out in the comments, his paper with David Gepner gives a very general approach to your wish list. However to make it so general that it applies to arbitrary monoidal $(\infty,1) …
Chris Schommer-Pries's user avatar
18 votes
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What is the free symmetric monoidal $\infty$-category on one object?

Yes, it is the same as $\mathbb{F}$. As John Baez points out, it is the same as the free symmetric monoidal $\infty$-groupoid on one object. (This can also be seen by playing around with the adjoints …
Chris Schommer-Pries's user avatar