  The purpose of this paper is two-fold. In Part 1 we introduce a new theory of
operadic categories and their operads. This theory is, in our opinion, of an
independent value.
  In Part 2 we use this new theory together with our previous results to prove
that multiplicative 1-operads in duoidal categories admit, under some mild
conditions on the underlying monoidal category, natural actions of contractible
2-operads. The result of D. Tamarkin on the structure of dg-categories, as well
as the classical Deligne conjecture for the Hochschild cohomology, is a
particular case of this statement.
