A category over an operad is the horizontal categorification of an algebra over an operad. It is like an enriched category in which the composition operation is not necessarily binary, but parameterized by the operad.
a set/class/whatever , called the set of objects of ;
for each pair an object , called the object of morphisms from to in ;
for each natural number and each sequence of objects of a morphism
called the -ary composition operation;
such that the composition operations satisfy the obvious compatibility conditions with the operad composition operation, directly analogous to those for -algebras.
We may regard the operad as a dg-operad i.e. an operad in the category of cochain complexes. As such, has a resolution, the A-infinity-operad operad . An -category is an A-infinity-category (see there).
A class of definitions of infinity-categories is operadic in this sense, or in a generalization thereof. See