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 $R_{-}$.

• Joyal's category $\Theta$ is also a Reedy category.

• Many very small categories of diagram shapes are Reedy categories, such as $(\cdot \to \cdot \to \dots )$, or $(\cdot \leftarrow \cdot \to \cdot )$, or $(\cdot \rightrightarrows \cdot )$. This is of importance for the construction of homotopy limits and colimits over such diagram shapes.

The simplex category

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.

The Reedy category structure on $\Delta$ is a follows

• a map $[k]\to [n]$ is in ${\Delta }_{+}$ precisely if it is injective;

• a map $[n]\to [k]$ is in ${\Delta }_{-}$ precisely if it is surjective.

(…)

Related notions

Direct and inverse categories

A Reedy category in which $R_{-}$ contains only identities is called a direct category; the factorization axiom then says simply that $R=R_{+}$. Similarly, if $R_{+}$ contains only identities it is said to be an inverse category.

Any ordinal is of course a direct category, and so is the subcategory $R_{+}$ of any Reedy category considered as a category in its own right. This amounts to “discarding the degeneracies” in a shape category. In some examples there are no degeneracies to begin with, such as the category of opetopes; thus these are naturally direct categories.

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 concept of

• generalized Reedy category,

due to Clemens Berger and Ieke Moerdijk, avoids these problems.

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.