Spatial refinements of chain complexes in the context of Khovanov homology

I will describe general strategies for spatial refinements of some naturally occurring chain complexes. Then I will concentrate on a specific chain complex: the Khovanov chain complex associated to a link diagram, and describe a specific strategy for constructing a CW complex whose cellular chain complex is the Khovanov complex. This work is joint with Robert Lipshitz, and parts are also joint with Tyler Lawson.