representation, ∞-representation?
symmetric monoidal (∞,1)-category of spectra
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
may equivalently be thought of, via the homotopy theory of ∞-groups, as part of the long homotopy fiber sequence
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
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
where now $c_{p+2}$ is the classifying $(p+2)$-cocyle.
This concept makese sense not just for ∞-groups in any (∞,1)-topos, but also for instance for (super) L-∞ algebras.
Last revised on August 1, 2018 at 04:29:10. See the history of this page for a list of all contributions to it.