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 It was introduced by Christopher Reedy in his unpublished manuscript.[1]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

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

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This section needs expansion. You can help by adding to it. (April 2025)

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## Eilenberg–Zilber category

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

## References

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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

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* 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

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

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