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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 11th 2011
• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeFeb 12th 2011

You defined “has adjoints” twice. It seems to me that the first should have been for $1$-morphisms and I changed it accordingly.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 12th 2011

Something I may have asked before, but we seem to deal with $\infty$-categories with various properties assigned to certain intervals of $n$-morphisms; no property, has duals, is invertible, is trivial. So we may say ’invertible above $n$’, or ’trivial above $n + k$ and below $k$ perhaps with duals between. How far do they ever get combined? I see we have k-tuply monoidal (n,r)-category. Do they crop up in nature? How about a k-tuply monoidal (infinity,n)-category with duals?

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeFeb 13th 2011

I see we have k-tuply monoidal (n,r)-category. Do they crop up in nature?

I mostly only wrote that to record the insight that the concept still makes sense when $r = -1$ (giving the concept of $k$-tuply groupal $n$-groupoid).

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 13th 2011

$k$-tuply monoidal $(n,r)$-categories certainly crop up in nature! How could they not?