Operadic categories and décalage

Batanin and Markl's operadic categories are categories in which each map is endowed with a finite collection of “abstract fibres”—also objects of the same category—subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the décalage comonad D on small categories. A simple case involves unary operadic categories—ones wherein each map has exactly one abstract fibre—which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monad D˜ induced on CatD by the forgetful–cofree adjunction. A similar description is found for general operadic categories arising out of a corresponding analysis that starts from a “modified décalage” comonad Dm on the arrow category Cat2.