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1. • CommentRowNumber1.
   • CommentAuthorjamievicary
   • CommentTimeJun 12th 2011
   • (edited Jun 12th 2011)
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   Author: jamievicary
   Format: MarkdownItexWhat's the correct higher-categorical jargon for expressing the categorification of the statement "unique up to unique isomorphism"? Surely there's a better way than "unique up to equivalence which is unique up to unique isomorphism". I suppose there's a homotopical answer to this -- "unique up to contractible equivalence" or something.

   What’s the correct higher-categorical jargon for expressing the categorification of the statement “unique up to unique isomorphism”? Surely there’s a better way than “unique up to equivalence which is unique up to unique isomorphism”. I suppose there’s a homotopical answer to this – “unique up to contractible equivalence” or something.

2. • CommentRowNumber2.
   • CommentAuthorFinnLawler
   • CommentTimeJun 12th 2011
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   Author: FinnLawler
   Format: MarkdownItexI don't know if it's correct in any sense, but I'd just say 'essentially unique' and rely on the context to determine exactly what that meant. Would it make sense to say, if you wanted to be precise, that a thing is _essentially $n$-unique_ (or something like that) if any other such thing is connected to it by an equivalence which is itself essentially $(n-1)$-unique (where $0$-unique means strictly unique, of course)?

   I don’t know if it’s correct in any sense, but I’d just say ’essentially unique’ and rely on the context to determine exactly what that meant.

   Would it make sense to say, if you wanted to be precise, that a thing is essentially $n$-unique (or something like that) if any other such thing is connected to it by an equivalence which is itself essentially $(n-1)$-unique (where $0$-unique means strictly unique, of course)?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 12th 2011
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   Author: Mike Shulman
   Format: MarkdownItexI'd say "essentially unique" or "unique up to a contractible space of choices" or (if the homotopical/higher context were understood) simply "unique".

   I’d say “essentially unique” or “unique up to a contractible space of choices” or (if the homotopical/higher context were understood) simply “unique”.

4. • CommentRowNumber4.
   • CommentAuthorjamievicary
   • CommentTimeJun 13th 2011
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   Author: jamievicary
   Format: MarkdownItexThanks guys. I think the most sensible route is to define "essentially unique", as it's a short enough phrase, and use that. I like the idea of $n$-uniqueness!

   Thanks guys. I think the most sensible route is to define “essentially unique”, as it’s a short enough phrase, and use that. I like the idea of $n$-uniqueness!

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJun 14th 2011
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   Author: zskoda
   Format: MarkdownItexThe words "essentially unique" would at least to me mean that it is unique up to an isomorphism, and not up to unique isomorphism (geometers in 1-categorical case distinguish unique up to iso and "uniquely unique" what is what you want: the Grothendieck school likes that repetition of the word unique as a reminder).

   The words “essentially unique” would at least to me mean that it is unique up to an isomorphism, and not up to unique isomorphism (geometers in 1-categorical case distinguish unique up to iso and “uniquely unique” what is what you want: the Grothendieck school likes that repetition of the word unique as a reminder).