nLab Eilenberg-Zilber category

Eilenberg-Zilber categories

Eilenberg-Zilber categories

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 RR equipped with a function d:ob(R)d \colon ob(R) \to \mathbb{N} such that

  1. For f:xyf \colon x\to y a morphism of RR:

    1. If ff is an isomorphism, then deg(x)=deg(y)deg(x)=deg(y).

    2. If ff is a noninvertible monomorphism, then deg(x)<deg(y)deg(x)\lt deg(y).

    3. If ff is a noninvertible split epimorphism, then deg(x)>deg(y)deg(x) \gt deg(y).

  2. Every morphism factors as a split epimorphism followed by a monomorphism.

  3. Any pair of split epimorphisms in RR 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][R^{op},Set] is an epimorphism, condition (2) in Def. says that the (epi, mono) factorization system of [R op,Set][R^{op},Set] restricts to RR via the Yoneda embedding, while condition (3) says that the representables are closed in [R op,Set][R^{op},Set] under pushouts of pairs of epimorphisms.

Properties

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

Any element of a presheaf on an EZ-category RR 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 Δ\Delta is an Eilenberg–Zilber category.

Example

The wreath product of Δ\Delta and an EZ-category (also known as the Θ\Theta-construction) is again an EZ-category (Bergner–Rezk, Proposition 4.3). In particular, Joyal’s category Θ n\Theta_n is an EZ-category for all n0n\ge0.

Example

Segal's category Γ\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 Λ\Lambda and the category of trees Ω\Omega are EZ-categories (Berger & Moerdijk, Examples 6.8).

Example

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

Example

The category of cubes QQ (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 min-connections (Isaacson, Definition 3.4, Proposition 3.11) is an EZ-category (Isaacson, Proposition 4.4).

References

Last revised on December 4, 2023 at 21:31:00. See the history of this page for a list of all contributions to it.