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I have added to all Segal space-related entries, as well as to the Example section at category object in an (infinity,1)-category statements like
a pre-category object in $\infty Grpd$ is called a Segal space;
a connected pre-category object in $\infty Grpd$ is called a reduced Segal space;
a category object in $\infty Grpd$ is called a complete Segal space.
an category object in $Cat(Cat(\cdots Cat(\infty Grpd)))$ is called an n-fold complete Segal space;
That list can be further expanded. But I have to quit now.
prompted by Mike’s remark here I added to Segal space a section Examples – in 1Grpd with a remark on how a Segal space in $1Grpd \hookrightarrow \infty Grpd$ induces a 2-category equipped with proarrows.
It’s not very polished and still sort of incomplete. But I need to quit now.
I have now also spelled out the converse construction in Examples – From a category, spelling out how for $\mathcal{C}$ a category and $p \colon \mathcal{K} \to \mathcal{C}$ a functor out of a groupoid, the “$n$-fold comma catgeory” construction $X_n \coloneqq p^{/^n}$ yields a Segal space $X_\bullet$, which is complete if $p$ is the core inclusion.
So far the writeup is a bit rambling, I am just writing stuff out as I go along. Eventually when I have a more leisure I should go an polish this. There should be, I suppose, a statement and proof that
(Here $X_\bullet$ is “$k$-coskeletal” if $X_n \to X^{\partial \Delta^n}$ is a 1-monomorphism for all $n \geq k+1$).
This is very nice, thanks!
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