On an extension of the notion of Reedy category

We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occurring in topology such as Segal’s category Γ, or the total category of a crossed simplicial group such as Connes’ cyclic category Λ. For any generalized Reedy category R and any cofibrantly generated model category , the functor category R is shown to carry a canonical model structure of Reedy type