nLab Eilenberg-Zilber category

Idea

An Eilenberg-Zilber category is a special sort of generalized Reedy category for which degeneracies behave particularly well.

Definition

An Eilenberg-Zilber category (or EZ-category) is a small category $R$ equipped with a function $d:ob(R) \to \mathbb{N}$ such that

1. For $f:x\to y$,
   1. If $f$ is an isomorphism, then $deg(x)=deg(y)$.
   2. If $f$ is a noninvertible monomorphism, then $deg(x)\lt deg(y)$.
   3. If $f$ is a noninvertible split epimorphism, then $deg(x) \gt deg(y)$.
2. Every morphism factors as a split epimorphism followed by a monomorphism.
3. Any pair of split epimorphisms in $R$ has an absolute pushout.

Since a morphism is a split epimorphism if and only if its image in the presheaf category $[R^{op},Set]$ is an epimorphism, condition (2) says that the (epi, mono) factorization system of $[R^{op},Set]$ restricts to $R$ via the Yoneda embedding, while condition (3) says that the representables are closed in $[R^{op},Set]$ under pushouts of pairs of epimorphisms.

Properties

Any EZ-category is a generalized Reedy category where $R^+$ and $R^-$ are the monomorphisms and the split epimorphisms, respectively. Moreover, $R^{op}$ is also a generalized Reedy category where the definitions of $R^+$ and $R^-$ are reversed. However, the generalized Reedy model structures on contravariant functors (corresponding to the generalized Reedy structure on $R^{op}$) are generally better-behaved.

Any element of a presheaf on an EZ-category $R$ 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.

References

• Clemens Berger, Ieke Moerdijk, On an extension of the notion of Reedy category, Mathematische Zeitschrift, 269, 2011 (arXiv:0809.3341, doi:10.1007/s00209-010-0770-x)
• Samuel Isaacson, Symmetric cubical sets, Journal of Pure and Applied Algebra, 215, 2011 (arXiv:0910.4948, doi:10.1016/j.jpaa.2010.08.001)
• Julia Bergner, Charles Rezk, Reedy categories and the Θ-construction, Mathematische Zeitschrift, 274, 2013 (arXiv:1110.1066, doi:10.1007/s00209-012-1082-0)
• Denis-Charles Cisinski, Higher Categories and homotopical algebra, Cambridge University Press, 2019, doi:10.1017/9781108588737)