Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
polished and expanded somewhat the entry groupoid object in an (infinity,1)-category
added to groupoid object in an (infinity,1)-category a subsecton with a remark on the notion of $(\infty,1)$-quotients / homotopy quotients.
finally added the central theorem about delooping in an $\infty$-topos: to a new section Delooping
I have added to the References at groupoid object in an (∞,1)-category the items
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?
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.
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”.
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.
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.
Urs: the link in #8 does not work as you made a typo. suspension object
Fixed.
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.
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.
Is there a reason for your use of the letter $\mathbf{B}$?
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 $B G$.
1 to 14 of 14