nLab higher central extension

Idea

In higher algebra a higher central extension of some algebraic object should be a higher analog of a central extension, hence a higher extension? by something in the higher analog of the center. What exactly “higher center” should mean is subject to some choices, see for instance center of an ∞-group.

But there is an easily identified sub-class of higher extensions that should definitely count as higher central extensions:

For $G$ a group, an ordinary central extension

$$1 \to A \longrightarrow \widehat G \longrightarrow G \to 1$$

may equivalently be thought of, via the homotopy theory of ∞-groups, as part of the long homotopy fiber sequence

$$\array{ A &\longrightarrow& \widehat G \\ && \Big\downarrow \\ && G &\overset{c_2}{\longrightarrow}& B A }$$

where $B A$ is the delooping 2-group of $A$, and where $c_2 \;\colon\; G \longrightarrow B A$ is the cocycle which classifies the central extension.

Equivalently, under further delooping this is the long homotopy fiber sequence of the form

$$\array{ B A &\longrightarrow& B \widehat G \\ && \Big\downarrow \\ && B G &\overset{c_2}{\longrightarrow}& B^2 A }$$

which makes explicitl that $c_2$ is a 2-cocycle.

Phrased this way it is clear that a concept of higher central extension of an ∞-group $G$ should subsume at least those long homotopy fiber sequence of the form

$$\array{ B^{p+1} A &\longrightarrow& B \widehat G \\ && \Big\downarrow \\ && B G &\overset{c_{p+2}}{\longrightarrow}& B^{p+2} A }$$

where now $c_{p+2}$ is the classifying $(p+2)$-cocycle.

This concept makese sense not just for ∞-groups in any (∞,1)-topos, but also for instance for (super) L-∞ algebras.

Examples

• higher central ∞-group extension

• L-∞ algebra extension

Related concepts

• universal higher central extension

• infinity-Lie algebra cohomology

References

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, section 4.3 of Principal infinity-bundles - General theory, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015) (arXiv:1207.0248)