  We define reduced colored sl(N) link homologies and use deformation spectral
sequences to characterize their dependence on color and rank. We then define
reduced colored HOMFLY-PT homologies and prove that they arise as large N
limits of sl(N) homologies. Together, these results allow proofs of many
aspects of the physically conjectured structure of the family of type A link
homologies. In particular, we verify a conjecture of Gorsky, Gukov and
Sto\v{s}i\'c about the growth of colored HOMFLY-PT homologies.
