Operadic categories and décalage

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