The Khovanov chain complex is a categorification of the Jones polynomial and is built using a functor from Kauffman’s cube of resolutions to Abelian groups.
By lifting this functor to the Burnside category, one can construct a CW complex whose reduced cellular chain complex agrees with the Khovanov complex.
This produces a Khovanov homotopy type whose reduced homology is Khovanov homology. I will present a general outline of this construction, starting with
Khovanov’s functor. This work is joint with Tyler Lawson and Robert Lipshitz.