nLab Eilenberg-Zilber category

Idea

Eilenberg-Zilber categories are a special sort of generalized Reedy categories for which degeneracy maps behave particularly well.

Definition

Definition

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

1. For $f \colon x\to y$ a morphism of $R$:

   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.

Remark

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) in Def. 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.

Examples

Example

The category of simplices $\Delta$ is an Eilenberg–Zilber category.

Example

The wreath product of $\Delta$ and an EZ-category (also known as the $\Theta$-construction) is again an EZ-category (Bergner–Rezk, Proposition 4.3). In particular, Joyal’s category $\Theta_n$ is an EZ-category for all $n\ge0$.

Example

Segal's category $\Gamma$ (used to define Gamma-spaces) is an EZ-category (Berger & Moerdijk 2011, Examples 6.8).

Example

The category of symmetric simplices (inhabited finite sets and their maps) is an EZ-category (Berger & Moerdijk, Examples 6.8).

Example

The cyclic category $\Lambda$ and the category of trees $\Omega$ are EZ-categories (Berger & Moerdijk, Examples 6.8).

Example

More generally, the total category $\mathbf{R}G$ of a crossed group $G$ on an EZ-category $\mathbf{R}$ whose underlying Reedy category is strict is itself an EZ-category (Berger & Moerdijk, Examples 6.8).

Example

The category of cubes $Q$ (generated by faces and degeneracies, without connections, symmetries, reversals, or diagonals) is an EZ-category (Isaacson 2010, Proposition 4.4).

Example

The category of symmetric cubes with (non-commutative) min-connections (Isaacson, Definition 3.4, Proposition 3.11) is an EZ-category (Isaacson, Proposition 4.4). The same is true for the category with commutative min-connections (Campion, Corollary 7.10).

Example

The cartesian cube category is an EZ-category (Campion, Theorem 8.12(1)).

Eilenberg–Zilber presheaves

Even if $R$ is a non-EZ generalized Reedy category, we can consider the following class of presheaves.

Definition

(Duliński 2026, Definition 3.4) Let $R$ be a generalized Reedy category. A presheaf $X$ on $R$ is an EZ-presheaf if every element of $X$ is a degeneracy of a unique nondegenerate element.

If $R$ is an EZ-category, then all presheaves on $X$ are EZ-presheaves. Duliński (Duliński 2026) studies the class of EZ-presheaves in the case where $R$ is a strict Reedy category. Under the following degeneracy-forking condition on $R$, relative cell complex decompositions are available for monomorphisms between EZ-presheaves.

Definition

(Duliński 2026, Definition 3.11, rephrased) A Reedy category $R$ is degeneracy-forking if every $\sigma \colon y \to z$ in $R^-$ for which $R^-(-,\sigma) \colon R^-(-,y) \to R^-(-,z)$ is a monomorphism is an identity.

Proposition

(Duliński 2026, Lemma 3.17) If $R$ is a degeneracy-forking Reedy category, then every monomorphism between EZ-presheaves on $R$ is a relative cell complex of boundary inclusions.

Proposition

(Duliński 2026, Corollary 3.21) If $R$ is a degeneracy-forking Reedy category, then $X \in [R^{op},Set]$ is an EZ-presheaf if and only if $0 \to X$ is a cell complex of boundary inclusions.

When $R^-$ has binary pushouts, the EZ-presheaves are exactly the presheaves that are right orthogonal to the comparison map $R(-,y) \sqcup_{R(-,x)} R(-,z) \to R(-,y \sqcup_x z)$ for every span $y \leftarrow x \rightarrow z$ in $R^-$ (Duliński 2026, Theorem 3.31). As a corollary:

Proposition

(Duliński 2026, Corollary 3.32) If $R$ is a Reedy category and $R^-$ has binary pushouts, then the full subcategory $[R^{op},Set]_{EZ} \hookrightarrow [R^{op},Set]$ of EZ-presheaves on $R$ is a reflective subcategory. In particular, $[R^{op},Set]_{EZ}$ is a locally presentable 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)

• Tim Campion, Cubical sites as Eilenberg-Zilber categories, 2023, arXiv:2303.06206

• Wojciech Zbigniew Duliński?, An (∞,0)-model structure on Eilenberg–Zilber opetopic sets, Mathematical Structures in Computer Science 36 (2026) 1–50 [doi:10.1017/S0960129526100620]