Reedy category

In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. A prototypical example is the simplex category or its opposite. It was introduced by Christopher Reedy in his unpublished manuscript.^[1]

Definition

A Reedy category consists of the following data: a category R, two wide (lluf) subcategories ${\displaystyle R_{-},R_{+}}$ and a functorial factorization of each map into a map in ${\displaystyle R_{-}}$ followed by a map in ${\displaystyle R_{+}}$ that are subject to the condition: for some total preordering (degree), the nonidentity maps in ${\displaystyle R_{-},R_{+}}$ lower or raise degrees.^[2]

Note some authors such as nlab require each factorization to be unique.^[3][4]

Reedy model structure

A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category.

Eilenberg–Zilber category

An Eilenberg–Zilber category is a variant of a Reedy category.

References

1. Reedy's manuscript can be found at https://math.mit.edu/~psh/
2. Barwick 2007, Definition 1.6.
3. "Reedy category". nLab.
4. "The definition of Reedy category". mathoverflow.

Literature

• Barwick, Clark (2007), On Reedy Model Categories, arXiv:0708.2832
• Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1-108-47320-0.
• 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)
• Tim Campion, Cubical sites as Eilenberg-Zilber categories, 2023, arXiv:2303.06206

Further reading

• Reedy category, Reedy model structure and Eilenberg-Zilber category at the nLab
• http://pantodon.jp/index.rb?body=Reedy_category in Japanese