Eilenberg-Zilber categories are a special sort of generalized Reedy categories for which degeneracy maps behave particularly well.
An Eilenberg-Zilber category (or EZ-category) is a small category equipped with a function such that
For a morphism of :
If is an isomorphism, then .
If is a noninvertible monomorphism, then .
If is a noninvertible split epimorphism, then .
Every morphism factors as a split epimorphism followed by a monomorphism.
Since a morphism is a split epimorphism if and only if its image in the presheaf category is an epimorphism, condition (2) in Def. 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.
The category of simplices is an Eilenberg–Zilber category.
The wreath product of and an EZ-category (also known as the -construction) is again an EZ-category (Bergner–Rezk, Proposition 4.3). In particular, Joyal’s category is an EZ-category for all .
Segal's category (used to define Gamma-spaces) is an EZ-category (Berger & Moerdijk 2011, Examples 6.8).
The category of symmetric simplices? (inhabited finite sets and their maps) is an EZ-category (Berger & Moerdijk, Examples 6.8).
The cyclic category and the category of trees are EZ-categories (Berger & Moerdijk, Examples 6.8).
More generally, the total category of a crossed group on an EZ-category whose underlying Reedy category is strict is itself an EZ-category (Berger & Moerdijk, Examples 6.8).
The category of cubes (generated by faces and degeneracies, without connections, symmetries, reversals, or diagonals) is an EZ-category (Isaacson 2010, Proposition 4.4).
The category of symmetric cubes with min-connections (Isaacson, Definition 3.4, Proposition 3.11) is an EZ-category (Isaacson, Proposition 4.4).
The cartesian cube category is an EZ-category (Campion, Theorem 8.12(1)).
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)
Tim Campion, Cubical sites as Eilenberg-Zilber categories, 2023, arXiv:2303.06206
Last revised on September 29, 2024 at 20:45:24. See the history of this page for a list of all contributions to it.