A 2-category of chronological cobordisms and odd Khovanov homology
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.