nLab
Reedy category

Idea

A Reedy category is a category $R$ equipped with a structure enabling the inductive construction of diagrams and natural transformations of shape $R$.

The most important consequence of a Reedy structure on $R$ is the existence of a model structure on the functor category $M^R$ whenever $M$ is a model category (no extra hypotheses on $M$ are required): the

• Reedy model structure .

Definition

A Reedy category is a category $R$ equipped with two lluf subcategories $R_{+}$ and $R_{-}$ and a function $d:\mathrm{ob}(R)\to \alpha$ called degree, where $\alpha$ is an ordinal number, such that:

• Every nonidentity morphism in $R_{+}$ raises degree,
• Every nonidentity morphism in $R_{-}$ lowers degree, and
• Every morphism $f$ in $R$ factors uniquely as a map in $R_{-}$ followed by a map in $R_{+}$.

Examples

• Any ordinal $\alpha$, considered as a poset and hence a category, is a Reedy category with ${\alpha }_{+}=\alpha$, ${\alpha }_{-}$ the discrete category on $\mathrm{ob}(\alpha )$, and $d$ the identity.

• The opposite of any Reedy category is a Reedy category; simply exchange $R_{+}$ and $R_{-}$.

• The prototypical examples of Reedy categories are the simplex category $\Delta$ and its opposite ${\Delta }^{\mathrm{op}}$. More generally, for any simplicial set $X$, its category of simplices $\Delta /X$ is a Reedy category.

Generalizations

generalized Reedy category

One problem with the notion of Reedy category is that it is evil: it is not invariant under equivalence of categories. It’s not hard to see that any Reedy category is necessarily skeletal. In fact, it’s even worse: no Reedy category can have any nonidentity isomorphisms! This is problematic for many $\Delta$-like categories such as the category of cycles, Segal’s category $\Gamma$, the tree category $\Omega$, and so on.

The following generalization of the notion of Reedy category, due to Ieke Moerdijk and Clemens Berger, avoids these problems and maintains the truth of the Theorem relating Reedy categories and model structures from Reedy model structure. Define a generalized Reedy category to be a category $R$ with the same structure $R_{+}$, $R_{-}$, and $d$ as before, but now such that

• every non-isomorphism in $R_{+}$ raises degree,
• every non-isomorphism in $R_{-}$ lowers degree,
• every morphism $f$ factors as a map in $R_{-}$ followed by a map in $R_{+}$, uniquely up to isomorphism, and
• If $f\in R_{-}$ and $\theta$ is an isomorphism such that $\theta f=f$, then $\theta =1$ (isomorphisms see the maps in $R_{-}$ as epis).

The last condition is not self-dual, but has an obvious dual version.

enriched Reedy category

There is also a generalization of the notion of Reedy category to the context of enriched category theory: this is an enriched Reedy category.