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Algebraic structures with multiple copies of a given type of operationsinterrelated by various compatibility conditions have long being studied inmathematics and mathematical physics. They are broadly referred as linearlycompatible, matching, and totally compatible structures. This paper gives aunified approach to these structures in the context of operads. We firstgeneralize the process of polarization for polynomials in invariant theory tothe one for operads, leading to the general notion of linearly compatibleoperads. Refining the polarization by partitioning it into foliations, weobtain a notion of matching operads consolidating those appeared recently fromapplications of regularity structures, and Volterra integral equations.Distinguished among them is the leveled matching compatibility, which is uniquewith respect to a fix ordering of the vertices in tree monomials. Equating allmatching compatibilities of a given operad leads to the totally compatibleoperad of this operad.  For unary/binary quadratic operads, the linear compatibility and the totalcompatibility are in Koszul dual to each other, and there is a Koszulself-duality among the matching compatibilities. In particular, the leveledmatching compatibility is Koszul self-dual. For binary quadratic operads, thesethree compatible operads can also be obtained by taking Manin black and whiteproducts. For finitely generated binary quadratic operad, we prove that theKoszulity is preserved under taking each of the linear and totalcompatibilities, as well as the leveled matching compatibility.

Compatible structures of operads by polarization, their Koszul duality and Manin products