equivalences in/of $(\infty,1)$-categories
An ∞-group is a group object in ∞Grpd.
Equivalently (by the delooping hypothesis) it is a pointed connected $\infty$-groupoid.
Under the identification of ∞Grpd with Top this is known as an $A_\infty$-space, for instance.
An $\infty$-Lie group is accordingly a group object in ∞-Lie groupoids. And so on.
For details see groupoid object in an (∞,1)-category.
By
$\infty$-group, braided ∞-group
(∞,1)-operad | ∞-algebra | grouplike version | in Top | generally | |
---|---|---|---|---|---|
A-∞ operad | A-∞ algebra | ∞-group | A-∞ space, e.g. loop space | loop space object | |
E-k operad? | E-k algebra | k-monoidal ∞-group | iterated loop space | iterated loop space object | |
E-∞ operad | E-∞ algebra | abelian ∞-group | E-∞ space, if grouplike: infinite loop space $\simeq$ ∞-space | infinite loop space object | |
$\simeq$ connective spectrum | $\simeq$ connective spectrum object | ||||
stabilization | spectrum | spectrum object |
A standard textbook reference on $\infty$-groups in the classical model structure on simplicial sets is
Group objects in (infinity,1)-categories are the topic of
Model category presentations of group(oid) objects in $\infty Grpd$ by groupoidal complete Segal spaces are discussed in
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)
Discussion from the point of view of category objects in an (∞,1)-category is in
The homotopy theory of $\infty$-groups that are n-connected and r-truncated for $r \leq n$ is discussed in
Discussion in homotopy type theory is in
For more see the references at infinity-action.
Last revised on February 20, 2018 at 07:36:49. See the history of this page for a list of all contributions to it.