  We show that a functor category whose domain is a colored category is a
topos.The topos structure enables us to introduce cohomology of colored
categories including quasi-schemoids. If the given colored category arises from
an association scheme, then the cohomology coincides with the group cohomology
of the factor scheme by the thin residue. Moreover, it is shown that the
cohomology of a colored category relates to the standard representation of an
association scheme via the Leray spectral sequence.
