  In this paper, we discuss degree 0 crossing change on Khovanov homology in
terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology,
we introduce a sum of cobordisms that yields a morphism on complexes of two
diagrams of crossing change, which we call the "genus-one morphism." It is
proved that the morphism is invariant under the moves of double points in
tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking
iterated mapping cones, we obtain an invariant for singular tangles that
extending sl(2) tangle homology; examples include Lee homology, Bar-Natan
homology, and Naot's universal Khovanov homology as well as Khovanov homology
with arbitrary coefficients. We also verify that the invariant satisfies
categorified analogues of Vassiliev skein relation and the FI relation.
