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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 21st 2012
• (edited May 21st 2012)

I have moved a Properties-section to strict n-category, taken from (infinity,n)-category (where it serves as a preliminary), which collects the statements from section 2 of Barwick&Schommer-Pries (see the refereces there).

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeMay 21st 2012

I see that they define a suspension of $n$-category as certain $n+1$-category with 2 objects. But the tradition (especially used by Baez) is that one calls suspension of $C$ as the $n+1$-category with only one object and with $C$-worth of endomorphisms of that object. Why the change of terminology is warranted ? Any comments ?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 21st 2012

It’s just a word. Feel free to invent another word.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeMay 21st 2012

So what will be your usage – which one will you call a suspension ?

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMay 21st 2012

IMHO what you call the “traditional” usage is wrong. It only make sense when $C$ is monoidal, and according to the topological analogy, that construction should really be called the delooping. Delooping and suspension are equivalent in the stable world (spectra) but not in the unstable world. This point confused me for a while when I was first learning about higher categories, so I think we should eschew the use of “suspension” to mean “delooping”. The B-CP usage of “suspension” is better.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMay 21st 2012

If I had written that article I had tried to find some different word, to avoid overuse.

But I don’t think it matters since this word is not going to be used broadly, it’s just some word for convenience in those paragraphs where it appears. You can just call it “$\sigma$”. Or anything else. :-)

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeMay 21st 2012

Mike, that is about discussion which I was seeking for. Thanks, Urs, as well.