In this introduction for topologists, we explain the role that
extensions of L-infinity algebras by taking homotopy fibers plays in
physics. This first appeared with the work of physicists D'Auria and Fre
in 1982, but is beautifully captured by the "brane bouquet" of Fiorenza,
Sati and Schreiber which shows how physical objects such as "strings",
"D-branes" and "M-branes" can be classified by taking successive
homotopy fibers of an especially simple L-infinity algebra called the
"supertranslation algebra". We then conclude by describing our joint
work with Schreiber where we build the brane bouquet out of the homotopy
theory of an even simpler L-infinity algebra called the "superpoint".
