added these references on development of $\infty$-category theory internal to any (∞,1)-topos:

internal (∞,1)-Yoneda lemma:

- Louis Martini,
*Yoneda’s lemma for internal higher categories*, [arXiv:2103.17141]

internal (infinity,1)-limits and (infinity,1)-colimits:

- Louis Martini, Sebastian Wolf,
*Limits and colimits in internal higher category theory*[arXiv:2111.14495]

internal cocartesian fibrations and straightening functor:

- Louis Martini,
*Cocartesian fibrations and straightening internal to an ∞-topos*[arXiv:2204.00295]

internal presentable (∞,1)-categories:

- Louis Martini, Sebastian Wolf,
*Presentable categories internal to an ∞-topos*[arXiv:2209.05103]

Added reference to

- Louis Martini,
*Yoneda’s lemma for internal higher categories*, (arXiv:2103.17141)

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

]]>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.

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

]]>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.

Sometimes it is called “gaunt”.

]]>Sometimes it is called “rigid”.

]]>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).

]]>Interesting!

]]>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”.

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