related by the Dold-Kan correspondence
An elegant Reedy category is a Reedy category such that the following equivalent conditions hold
Both the following conditions hold:
For every monomorphism of presheaves on , every nondegenerate element of remains nondegenerate in .
Every element of a presheaf is a degeneracy of some nondegenerate element in a unique way.
In particular, if is elegant, then every codegeneracy map is a split epimorphism.
In particular, this implies that every -valued presheaf on an elegant Reedy category is Reedy cofibrant.
The simplex category is an elegant Reedy category.
Joyal’s disk categories are elegant Reedy categories.
Every direct category is a Reedy category with no degeneracies, hence trivially an elegant one.
If is any presheaf on an elegant Reedy category , then the opposite of its category of elements is again an elegant Reedy category. This is fairly easy to see from the fact that is equivalent to the slice category .
Every EZ-Reedy category? is elegant.
Note that unlike the notion of Reedy category, the notion of elegant Reedy category is not self-dual: if is elegant then will not generally be elegant.
Elegant Reedy categories are useful to model homotopy type theory.
Benno van den Berg and Ieke Moerdijk, W-types in homotopy type theory, PDF