This paper is devoted to the study of algebraic structures leading to link
homology theories. The originally used structures of Frobenius algebra and/or
TQFT are modified in two directions. First, we refine 2-dimensional cobordisms
by taking into account their embedding into the three space. Secondly, we
extend the underlying cobordism category to a 2-category, where the usual
relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is
called an extended quantum field theory (EQFT). We show that the Khovanov
homology, the nested Khovanov homology, extracted by Stroppel and Webster from
Seidel-Smith construction, and the odd Khovanov homology fit into this setting.
Moreover, we prove that any EQFT based on a Z_2-extension of the embedded
cobordism category which coincides with Khovanov after reducing the
coefficients modulo 2, gives rise to a link invariant homology theory
isomorphic to those of Khovanov.Comment: Lots of figure
