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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 9th 2012
   • (edited Nov 26th 2012)
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   Author: Urs
   Format: MarkdownItexI have added to all _[[Segal space]]_-related entries, as well as to the [Example section](http://ncatlab.org/nlab/show/category+object+in+an+%28infinity%2C1%29-category#OrdinaryInfinityCategories) 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2012
   • (edited Nov 26th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexprompted by Mike's remark [here](http://nforum.mathforge.org/discussion/3650/internal-infinity1category/?Focus=32171#Comment_32171) I added to [[Segal space]] a section _[Examples -- in 1Grpd](http://ncatlab.org/nlab/show/Segal+space#ExamplesInIGrpd)_ 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2012
   • (edited Nov 26th 2012)
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   Author: Urs
   Format: MarkdownItexI have now also spelled out the converse construction in [Examples -- From a category](http://ncatlab.org/nlab/show/Segal+space#ConstructionFromACategory), 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 * a 2-coskeletal Segal space $X_\bullet$ in $1Grpd$ is precisely the comma-Cech nerve of a functor $X_0 \to \mathcal{C}$ and is complete Segal precisely if this is the core inclusion of $\mathcal{C}$, and all 2-coskeletal Segal spaces in $1Grpd$ arise this way. (Here $X_\bullet$ is "$k$-coskeletal" if $X_n \to X^{\partial \Delta^n}$ is a [[1-monomorphism]] for all $n \geq k+1$).

   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

   • a 2-coskeletal Segal space $X_\bullet$ in $1Grpd$ is precisely the comma-Cech nerve of a functor $X_0 \to \mathcal{C}$ and is complete Segal precisely if this is the core inclusion of $\mathcal{C}$, and all 2-coskeletal Segal spaces in $1Grpd$ arise this way.

   (Here $X_\bullet$ is “$k$-coskeletal” if $X_n \to X^{\partial \Delta^n}$ is a 1-monomorphism for all $n \geq k+1$).

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeNov 27th 2012
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   Author: Mike Shulman
   Format: MarkdownItexThis is very nice, thanks!

   This is very nice, thanks!

5. • CommentRowNumber5.
   • CommentAuthornLab edit announcer
   • CommentTimeJun 2nd 2023
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   Author: nLab edit announcer
   Format: MarkdownItexAdded link to [[Segal type]] in "Related notions" Anna Kowalski <a href="https://ncatlab.org/nlab/revision/diff/Segal+space/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/Segal+space/21">v21</a>, <a href="https://ncatlab.org/nlab/show/Segal+space">current</a>

   Added link to Segal type in “Related notions”

   Anna Kowalski

   diff, v21, current