nLab
Eilenberg-Zilber category

Eilenberg-Zilber categories

Eilenberg-Zilber categories

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

  1. For f:xyf:x\to y,
    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.

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) 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 strurcture 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.

References

Last revised on May 13, 2021 at 18:42:52. See the history of this page for a list of all contributions to it.