Reedy category

Short description: Type of category in mathematics

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 It was introduced by Christopher Reedy in his unpublished manuscript.

Definition

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

Note some authors such as nlab require each factorization to be unique.

Eilenberg–Zilber category

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

Literature

• Barwick, Clark (2007), On Reedy Model Categories 
• Cisinski, Denis-Charles (2023) (in en). Higher Categories and Homotopical Algebra. Cambridge University Press. ISBN 978-1-108-47320-0. https://cisinski.app.uni-regensburg.de/CatLR.pdf. 
• 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