  We formulate a version of the integral Hodge conjecture for categories, prove
the conjecture for two-dimensional Calabi-Yau categories which are suitably
deformation equivalent to the derived category of a K3 or abelian surface, and
use this to deduce cases of the usual integral Hodge conjecture for varieties.
Along the way, we prove a version of the variational integral Hodge conjecture
for families of two-dimensional Calabi-Yau categories, as well as a general
smoothness result for relative moduli spaces of objects in such families. Our
machinery also has applications to the structure of intermediate Jacobians,
such as a criterion in terms of derived categories for when they split as a sum
of Jacobians of curves.
