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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexpolished and expanded somewhat the entry [[groupoid object in an (infinity,1)-category]]

   polished and expanded somewhat the entry groupoid object in an (infinity,1)-category

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 23rd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[groupoid object in an (infinity,1)-category]] a subsecton with a remark on the notion of $(\infty,1)$-quotients / homotopy quotients.

   added to groupoid object in an (infinity,1)-category a subsecton with a remark on the notion of $(\infty,1)$-quotients / homotopy quotients.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 29th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexfinally added the central theorem about delooping in an $\infty$-topos: to a new section <a href="http://ncatlab.org/nlab/show/groupoid+object+in+an+(infinity%2C1)-category#Delooping">Delooping</a>

   finally added the central theorem about delooping in an $\infty$-topos: to a new section Delooping

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2012
   • (edited Apr 13th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to the References at [[groupoid object in an (∞,1)-category]] the items * [[Julia Bergner]], _Adding inverses to diagrams encoding algebraic structures_, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. ([arXiv:0610291](http://arxiv.org/abs/math/0610291)) _Adding inverses to diagrams II: Invertible homotopy theories are spaces_, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. ([web](http://www.intlpress.com/hha/v10/n2/a9/), [arXiv:0710.2254](http://arxiv.org/abs/0710.2254)) wherein model category presentations of the $\infty$-categories of groupoid objects in $\infty Grpd$ are discussed. It seems sort of straightforward to generalize this to model categories presenting $\infty$-categories of groupoid objects in more general presentable $\infty$-categories / $\infty$-toposes. The groupoidal version of Segal space objects in model structures of simplicial (pre)sheaves. But: has it been written out? Is anyone aware of something?

   I have added to the References at groupoid object in an (∞,1)-category the items

   • Julia Bergner,

     Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. (arXiv:0610291)

     Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (web, arXiv:0710.2254)

   wherein model category presentations of the $\infty$-categories of groupoid objects in $\infty Grpd$ are discussed.

   It seems sort of straightforward to generalize this to model categories presenting $\infty$-categories of groupoid objects in more general presentable $\infty$-categories / $\infty$-toposes. The groupoidal version of Segal space objects in model structures of simplicial (pre)sheaves.

   But: has it been written out? Is anyone aware of something?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2012
   • (edited Apr 13th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn fact, even for the ambient $\infty$-category being $\infty Grpd$, a more thorough model theory presentation of the theory of groupoid objects would be desireable. The two articles mentioned above focus mostly on models for the actual objects of that $\infty$-category. It's section 3 of the second article that gives a genuine model for groupoid objects in $\infty Grpd$ ("invertible Segal spaces"). It would for instance be nice to have a Quillen equivalence form there to a model structure for effective epimorphisms in $\infty Grpd$. Things like that. I am wondering if this has been done in citable form somewhere, or if it still needs to be written out.

   In fact, even for the ambient $\infty$-category being $\infty Grpd$, a more thorough model theory presentation of the theory of groupoid objects would be desireable. The two articles mentioned above focus mostly on models for the actual objects of that $\infty$-category. It’s section 3 of the second article that gives a genuine model for groupoid objects in $\infty Grpd$ (“invertible Segal spaces”). It would for instance be nice to have a Quillen equivalence form there to a model structure for effective epimorphisms in $\infty Grpd$. Things like that. I am wondering if this has been done in citable form somewhere, or if it still needs to be written out.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 3rd 2012
   • (edited Dec 3rd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[groupoid object in an (infinity,1)-category]]_ a section _[Equivalent characterizations](http://ncatlab.org/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category#EquivalentCharacterizations)_ with some of those equivalent characterizations. Or rather, for the moment I have mostly concentrated on adding a little remark on how to translate from the "cone-style" conditions as they appear in [[HTT]] to the equivalent "powering-style"-conditions, as they appear in "[[(infinity,2)-Categories and the Goodwillie Calculus|I2CATGC]]".

   I have added to groupoid object in an (infinity,1)-category a section Equivalent characterizations with some of those equivalent characterizations.

   Or rather, for the moment I have mostly concentrated on adding a little remark on how to translate from the “cone-style” conditions as they appear in HTT to the equivalent “powering-style”-conditions, as they appear in “I2CATGC”.

7. • CommentRowNumber7.
   • CommentAuthorColin Tan
   • CommentTimeJul 21st 2014
   • (edited Jul 21st 2014)
   • PermaLink
   Author: Colin Tan
   Format: MarkdownItexThere is a certain ambiguity of denoting the (oo,1)-categorial equivalence from pointed connected object in a Grothendieck (oo,1)-topos to group objects in that topos by $\Omega$, namely see proposition 7 at [groupoid object in an (oo,1)-category](http://ncatlab.org/nlab/show/groupoid+object+in+an+%28infinity,1%29-category). This is because, for X a pointed connected object, the notation could mean either a pointed object internal to the said topos or a group object internal to the said topos. I was troubled by this ambiguity when writing up a proof at [suspension object](http://ncatlab.org/nlab/show/suspension+object) that, internal to a Grothendieck (oo,1)-topos, suspending is equivalent smashing with the classfying space of the integers. There, for G a group object, I use the notation $\Omega {\mathbf{B}} G$ to denote a pointed object. Taking the adjunction $(\Omega \vdash {\mathbf{B}})$ in the current notation per se, this notation $\Omega {\mathbf{B}} G$ ought to (by abstract nonsense) refer to a group object equivalent to $G$. One suggestion, if we follow Lurie and take the "complete Segal-space style" presentation of a group object as a simplicial object satisfying some conditions, then is to write the categorial equivalence as $\check{C}(*\to X):{\mathrm{PointedConn}}\to {\mathrm{Grp}}$, which sends a pointed connected object $X$ to the (underlying simplicial set) of the Cech nerve of the based map $*\to X$ from the terminal object.

   There is a certain ambiguity of denoting the (oo,1)-categorial equivalence from pointed connected object in a Grothendieck (oo,1)-topos to group objects in that topos by $\Omega$, namely see proposition 7 at groupoid object in an (oo,1)-category. This is because, for X a pointed connected object, the notation could mean either a pointed object internal to the said topos or a group object internal to the said topos.

   I was troubled by this ambiguity when writing up a proof at suspension object that, internal to a Grothendieck (oo,1)-topos, suspending is equivalent smashing with the classfying space of the integers. There, for G a group object, I use the notation $\Omega {\mathbf{B}} G$ to denote a pointed object. Taking the adjunction $(\Omega \vdash {\mathbf{B}})$ in the current notation per se, this notation $\Omega {\mathbf{B}} G$ ought to (by abstract nonsense) refer to a group object equivalent to $G$.

   One suggestion, if we follow Lurie and take the “complete Segal-space style” presentation of a group object as a simplicial object satisfying some conditions, then is to write the categorial equivalence as $\check{C}(*\to X):{\mathrm{PointedConn}}\to {\mathrm{Grp}}$, which sends a pointed connected object $X$ to the (underlying simplicial set) of the Cech nerve of the based map $*\to X$ from the terminal object.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2014
   • (edited Jul 21st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis is about whether to leave the forgetful functor from groups to pointed objects notationally implicit. By the way, over at [[suspension object]] you are using notation in both ways. First you say that $\Omega \mathbf{B}$ lands in pointed objects, but in the first proof you use $\Omega$ as landing in group objects. My suggestion is: say locally, eg in the entry on suspension, ecplicitly what the conventions are. There won't ever be consistent conventions across all nLab entries. Indeed, since you write F for the free group functor, it would be most ntural to call the forgetful functor just U, as usual, and not Omega B.

   This is about whether to leave the forgetful functor from groups to pointed objects notationally implicit.

   By the way, over at suspension object you are using notation in both ways. First you say that $\Omega \mathbf{B}$ lands in pointed objects, but in the first proof you use $\Omega$ as landing in group objects.

   My suggestion is: say locally, eg in the entry on suspension, ecplicitly what the conventions are. There won’t ever be consistent conventions across all nLab entries.

   Indeed, since you write F for the free group functor, it would be most ntural to call the forgetful functor just U, as usual, and not Omega B.

9. • CommentRowNumber9.
   • CommentAuthorTim_Porter
   • CommentTimeJul 21st 2014
   • (edited Jul 21st 2014)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexUrs: the link in #8 does not work as you made a typo. [[suspension object]]

   Urs: the link in #8 does not work as you made a typo. suspension object

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexFixed.

    Fixed.

11. • CommentRowNumber11.
    • CommentAuthorColin Tan
    • CommentTimeJul 21st 2014
    • (edited Jul 21st 2014)
    • PermaLink
    Author: Colin Tan
    Format: MarkdownItexThanks Urs, for pointing out the inconsistencies I made. There are really two right adjoint functors from group objects in question here. One is, following your suggestion, the functor $U:{\mathrm{Grp}}\to {\mathrm{Pointed}}$ which sends a group $G$ (regarded as a simplicial object) to $G_1$ (note that $G_1$ could possibly not be 0-connected). Its left adjoint is $F:{\mathrm{Pointed}}\to {\mathrm{Grp}}$ which sends a pointed object $X$ to the (underlying simplicial object) of the Cech nerve of $*\to \Sigma X$. The other is the functor ${\mathbf{B}}:{\mathrm{Grp}}\to{\mathrm{PointedConnected}}$ which sends a group $G$ to the colimit of $G$, regarded as a diagram over the simplex category. (The Lab calls this functor ${\mathbf{B}}$; In the 0-truncated case, this functor is usually called the nerve; Lurie calls it geometric realization $|-|$.) Its left adjoint ${\mathrm{PointedConnected}}\to{\mathrm{Grp}}$ sends a pointed connected object $X$ to the (underlying simplicial object) of the Cech nerve of $*\to X$. It is really the interplay of these two functors that proves the concretization of the suspension functor.

    Thanks Urs, for pointing out the inconsistencies I made.

    There are really two right adjoint functors from group objects in question here. One is, following your suggestion, the functor $U:{\mathrm{Grp}}\to {\mathrm{Pointed}}$ which sends a group $G$ (regarded as a simplicial object) to $G_1$ (note that $G_1$ could possibly not be 0-connected). Its left adjoint is $F:{\mathrm{Pointed}}\to {\mathrm{Grp}}$ which sends a pointed object $X$ to the (underlying simplicial object) of the Cech nerve of $*\to \Sigma X$. The other is the functor ${\mathbf{B}}:{\mathrm{Grp}}\to{\mathrm{PointedConnected}}$ which sends a group $G$ to the colimit of $G$, regarded as a diagram over the simplex category. (The Lab calls this functor ${\mathbf{B}}$; In the 0-truncated case, this functor is usually called the nerve; Lurie calls it geometric realization $|-|$.) Its left adjoint ${\mathrm{PointedConnected}}\to{\mathrm{Grp}}$ sends a pointed connected object $X$ to the (underlying simplicial object) of the Cech nerve of $*\to X$. It is really the interplay of these two functors that proves the concretization of the suspension functor.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes, and I am really fond of wring $\mathbf{B}$ and not writing "nerve" or "geometric realization" because the latter two are conceptually misleading or at best highly ambiguous, no matter how standard they may be.

    Yes, and I am really fond of wring $\mathbf{B}$ and not writing “nerve” or “geometric realization” because the latter two are conceptually misleading or at best highly ambiguous, no matter how standard they may be.

13. • CommentRowNumber13.
    • CommentAuthorColin Tan
    • CommentTimeAug 2nd 2014
    • PermaLink
    Author: Colin Tan
    Format: MarkdownItexIs there a reason for your use of the letter $\mathbf{B}$?

    Is there a reason for your use of the letter $\mathbf{B}$?

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2014
    • (edited Aug 10th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexSure, for $G$ a topological group regarded as an [[infinity-group]] via its underlying homotopy type, then the super-traditional [[classifying space]] construction $B G$ is the delooping of $G$ in the $\infty$-topos $\infty Grpd$. The boldface $\mathbf{B}$ is to denote delooping in any other $\infty$-topos. The boldface is to be suggestive of "delooping remembering additional structure" (fat structure). More precisely, if $\mathbf{H}$ is a cohesive $\infty$-topos then under suitable conditions the functor $\Pi : \mathbf{H}\longrightarrow \infty Grpd$ takes $\mathbf{B}G$ to $B G$.

    Sure, for $G$ a topological group regarded as an infinity-group via its underlying homotopy type, then the super-traditional classifying space construction $B G$ is the delooping of $G$ in the $\infty$-topos $\infty Grpd$. The boldface $\mathbf{B}$ is to denote delooping in any other $\infty$-topos. The boldface is to be suggestive of “delooping remembering additional structure” (fat structure). More precisely, if $\mathbf{H}$ is a cohesive $\infty$-topos then under suitable conditions the functor $\Pi : \mathbf{H}\longrightarrow \infty Grpd$ takes $\mathbf{B}G$ to $B G$.