The fundamental feature of Batanin-Markl's theory of operadic categories is that the objects under study are viewed as algebras over (generalized) operads in a speci c operadic category. Thus, for instance, ordinary operads arise as algebras over the terminal operad in the operadic category of rooted trees, modular operads are algebras over the terminal operad in the operadic category of genus-graded connected graphs, etc. Our aim is to construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads. According to general philosophy, algebras for these models describe strongly homotopy versions of the corresponding objects whose salient feature is the transfer property over weak homotopy equivalences. We describe our minimal models in terms of cellular chain complexes of convex polytopes, which delivers in an ingenious way the desired homology properties.