on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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 certain 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.
A Reedy category is a category $R$ equipped with two lluf subcategories $R_+$ and $R_-$ and a function $d:ob(R) \to \alpha$ called degree, where $\alpha$ is an ordinal number, such that:
Any ordinal $\alpha$, considered as a poset and hence a category, is a Reedy category with $\alpha_+=\alpha$, $\alpha_-$ the discrete category on $ob(\alpha)$, 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 $(\cdot\leftarrow \cdot\rightarrow\cdot)$, or $(\cdot\rightrightarrows\cdot)$. This is of importance for the construction of homotopy limits and colimits over such diagram shapes.
The prototypical examples of Reedy categories are the simplex category $\Delta$ and its opposite $\Delta^{op}$. 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.
(…)
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.
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
due to Clemens Berger and Ieke Moerdijk, avoids these problems. There is a similar generalization due to Denis-Charles Cisinski.
The notion of elegant Reedy category, introduced by Julie Bergner and Charles Rezk, is a restriction of the notion which captures the property that the Reedy model structure and injective model structure coincide. Several important Reedy categories are elegant, such as the $\Delta$ and $\Theta$.
There is also a generalization of the notion of Reedy category to the context of enriched category theory: this is an enriched Reedy category.
If $R$ is a direct category, then for any model category $M$ the colimit functor $\colim_R \colon M^R \to M$ is a left Quillen functor. However, there are non-direct Reedy categories with the same property, they are called Reedy categories with fibrant constants.
See the references at Reedy model structure