nLab
higher central extension

Contents

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 GG a group, an ordinary central extension

1AG^G1 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

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

where BAB A is the delooping 2-group of AA, and where c 2:GBAc_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

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

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

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

B p+1A BG^ BG c p+2 B p+2A \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+2c_{p+2} is the classifying (p+2)(p+2)-cocyle.

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

Examples

References

Last revised on August 1, 2018 at 04:29:10. See the history of this page for a list of all contributions to it.