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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeNov 28th 2012

It was pointed out to me today that in the very special case of internal (0,1)-category objects in Set, what we are calling a “pre-category” reduces to a preordered set, while adding the “univalence/Rezk-completeness” condition to make it a “category” promotes it to a partially ordered set. I feel like surely I knew that once, but if so, I had forgotten. It provides some extra weight behind this term “precategory”, especially since some category theorists like to say merely “ordered set” to mean “partially ordered set”.

• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeNov 28th 2012

Interesting!

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 28th 2012
• (edited Nov 28th 2012)

In general, the $sSet$-nerve of a category is complete Segal precisely if the only isomorphisms are identities (what’s the name again for such a category?).

I have added a paragraph on this to Segal space – Examples – In Set

(this could alternatively go to various other entries, but now I happen to have it there, and linked to from elsewhere).

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeNov 28th 2012

Sometimes it is called “rigid”.

1. Sometimes it is called “gaunt”.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 28th 2012
• (edited Nov 28th 2012)

Ah, of course. I knew that term of yours, I wrote all those notes on your article, after all. But I forgot. Thanks for reminding me. There is now an entry gaunt category, so that I shall never forget again.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeNov 29th 2012

Right, thanks. That’s better than “rigid”.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeDec 3rd 2012
• (edited Dec 3rd 2012)

I have been further expanding the Definition-section at category object in an (infinity,1)-category, adding a tad more details concerning proofs of some of the statements. But didn’t really get very far yet.

Also reorganized again slightly. I am afraid that in parts the notation is now slightly out of sync. I’ll get back to this later today.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeDec 4th 2012

I have edited still a bit further. This will probably be it for a while, unless I spot some urgent mistakes or omissions.