nLab
infinity-group

Context

Group Theory

(,1)(\infty,1)-Category theory

Contents

Definition

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 A_\infty-space, for instance.

An \infty-Lie group is accordingly a group object in ∞-Lie groupoids. And so on.

Properties

For details see groupoid object in an (∞,1)-category.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq Γ-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

References

The homotopy theory of \infty-groups that are n-connected and r-truncated for rnr \leq n 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

category: ∞-groupoid

Revised on April 1, 2014 11:06:04 by Urs Schreiber (89.204.155.26)