+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Group Theory +-- {: .hide} [[!include group theory - contents]] =-- #### $(\infty,1)$-Category theory +--{: .hide} [[!include quasi-category theory contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition An **∞-group** is a [[groupoid object in an (∞,1)-category|group object]] in [[∞Grpd]]. Equivalently (by the [[delooping hypothesis]]) it is a [[pointed object|pointed]] [[connected]] $\infty$-[[infinity-groupoid|groupoid]]. Under the [[homotopy hypothesis|identification]] of [[∞Grpd]] with [[Top]] this is known as a grouplike $A_\infty$-[[A-infinity-space|space]], for instance. An **$\infty$-Lie group** is accordingly a group object in [[∞-Lie groupoid]]s. And so on. ## Properties For details see [[groupoid object in an (∞,1)-category]]. ## Models By * [[model structure on simplicial groups]] * [[model structure on reduced simplicial sets]] ## Related concepts and Examples * [[group]] * [[2-group]], [[braided 2-group]] * [[n-group]], [[k-tuply groupal n-groupoid]] * [[pointed connected groupoid]] * **$\infty$-group**, [[braided ∞-group]] * [[looping and delooping]] * [[automorphism ∞-group]] * [[∞-group of bisections]] * [[∞-group of units]] * [[Picard ∞-group]] * [[Brauer ∞-group]] * [[p-compact group]] * [[center of an ∞-group]] * [[normal morphism of ∞-groups]] * [[stabilizer ∞-group]] * [[outer automorphism ∞-group]] * [[Eilenberg-MacLane object]], [[∞-gerbe]] * [[∞-group cohomology]] * [[∞-group extension]] * [[augmented ∞-group]] * [[∞-group completion]] * [[finite ∞-group]] * [[free infinity-group type]] * [[∞-action]] [[!include k-monoidal table]] ## References {#References} (For more see also the references at _[[infinity-action]]_.) A standard textbook reference on $\infty$-groups in the [[classical model structure on simplicial sets]] is * {#GoerssJardine} [[Paul Goerss]], [[Rick Jardine]], chapter V of _[[Simplicial homotopy theory]]_ [chapter V](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-5.dvi). [[groupoid object in an (infinity,1)-category|Group objects in (infinity,1)-categories]] are the topic of * {#Lurie} [[Jacob Lurie]], section 6.1.2 in _[[Higher Topos Theory]]_ Discussion from the point of view of [[category objects in an (∞,1)-category]] is in * {#Lurie2} [[Jacob Lurie]], _[[(∞,2)-Categories and the Goodwillie Calculus]]_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462)) The [[homotopy theory]] of $\infty$-groups that are [[n-connected]] and [[n-truncated|r-truncated]] for $n \leq r$ is discussed in * [[A.R. Garzón]], J.G. Miranda, _Serre homotopy theory in subcategories of simplicial groups_, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (<a href="https://doi.org/10.1016 /S0022-4049(98)00143-1">doi:10.1016/S0022-4049(98)00143-1</a>) Discussion of aspects of ordinary [[group theory]] in relation to $\infty$-group theory: * [[Roman Mikhailov]], *Homotopy patterns in group theory*, Proceedings of the [ICM 2022](https://icm2022.org) ([arXiv:2111.00737](https://arxiv.org/abs/2111.00737)) Discussion of [[infinity-groups|$\infty$-groups]] in [[homotopy type theory]]: * {#BuchholtzDoornRijke18} [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], *[[Higher Groups in Homotopy Type Theory]]*, LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (2018) 205-214 &lbrack;[arXiv:1802.04315](https://arxiv.org/abs/1802.04315), [doi:10.1145/3209108.3209150](https://doi.org/10.1145/3209108.3209150)&rbrack; [[!redirects infinity-groups]] [[!redirects infinity-group]] [[!redirects infinity-groups]] [[!redirects ∞-group]] [[!redirects ∞-groups]] [[!redirects pointed connected type]] [[!redirects pointed connected types]] category:∞-groupoid