  We create a framework for odd Khovanov homology in the spirit of Bar-Natan's
construction for the ordinary Khovanov homology. Namely, we express the cube of
resolutions of a link diagram as a diagram in a certain 2-category of
chronological cobordisms and show that it is 2-commutative: the composition of
2-morphisms along any 3-dimensional subcube is trivial. This allows us to
create a chain complex, whose homotopy type modulo certain relations is a link
invariant. Both the original and the odd Khovanov homology can be recovered
from this construction by applying certain strict 2-functors. We describe other
possible choices of functors, including the one that covers both homology
theories and another generalizing dotted cobordisms to the odd setting. Our
construction works as well for tangles and is conjectured to be functorial up
to sign with respect to tangle cobordisms.
