An Eilenberg-Zilber category is a special sort of generalized Reedy category for which degeneracies behave particularly well.
An Eilenberg-Zilber category (or EZ-category) is a small category equipped with a function such that
Since a morphism is a split epimorphism if and only if its image in the presheaf category is an epimorphism, condition (2) says that the (epi, mono) factorization system of restricts to via the Yoneda embedding, while condition (3) says that the representables are closed in under pushouts of pairs of epimorphisms.
Any EZ-category is a generalized Reedy category where and are the monomorphisms and the split epimorphisms, respectively. Moreover, is also a generalized Reedy category where the definitions of and are reversed. However, the generalized Reedy model structures on contravariant functors (corresponding to the generalized Reedy structure on ) are generally better-behaved.
Any element of a presheaf on an EZ-category is a degeneracy of a unique nondegenerate element.
If an EZ-category is also a strict Reedy category (i.e. contains no nonidentity isomorphisms), then it is an elegant Reedy category.
Last revised on April 5, 2023 at 09:01:46. See the history of this page for a list of all contributions to it.