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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”.
Interesting!
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).
Sometimes it is called “rigid”.
Sometimes it is called “gaunt”.
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.
Right, thanks. That’s better than “rigid”.
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.
I have edited still a bit further. This will probably be it for a while, unless I spot some urgent mistakes or omissions.
Added reference to
added these references on development of $\infty$-category theory internal to any (∞,1)-topos:
internal (∞,1)-Yoneda lemma:
internal (infinity,1)-limits and (infinity,1)-colimits:
internal cocartesian fibrations and straightening functor:
internal presentable (∞,1)-categories:
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