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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 11th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexstub for [[(infinity,n)-category with duals]]

   stub for (infinity,n)-category with duals

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 12th 2011
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   Author: TobyBartels
   Format: MarkdownItexYou defined "has adjoints" twice. It seems to me that the first should have been *for $1$-morphisms* and I changed it accordingly.

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

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 12th 2011
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   Author: David_Corfield
   Format: MarkdownItexSomething 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](http://ncatlab.org/nlab/show/k-tuply+monoidal+%28n%2Cr%29-category). Do they crop up in nature? How about a k-tuply monoidal (infinity,n)-category with duals?

   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?

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 13th 2011
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   Author: TobyBartels
   Format: MarkdownItex>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).

   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).

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 13th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex$k$-tuply monoidal $(n,r)$-categories certainly crop up in nature! How could they not?

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