Various versions of Khovanov chain complexes are built from functors from the cube to Abelian groups. By lifting these functors to the Burnside category, one can construct CW complexes whose cellular chain complexes agree with the Khovanov complexes. I will present a general outline of this construction, focusing specifically on the even and odd Khovanov homology. The even theory is joint with Tyler Lawson and Robert Lipshitz, and the odd theory is joint with Chris Scaduto and Matt Stoffregen.