related by the Dold-Kan correspondence
The most important consequence of a Reedy structure on is the existence of a certain model structure on the functor category whenever is a model category (no extra hypotheses on are required): the Reedy model structure.
The opposite of any Reedy category is a Reedy category; simply exchange and .
Joyal's category is also a Reedy category.
Many very small categories of diagram shapes are Reedy categories, such as , or , or . This is of importance for the construction of homotopy limits and colimits over such diagram shapes.
The Reedy category structure on is a follows
a map is in precisely if it is injective;
a map is in precisely if it is surjective.
A Reedy category in which contains only identities is called a direct category; the factorization axiom then says simply that . Similarly, if contains only identities it is said to be an inverse category.
Any ordinal is of course a direct category, and so is the subcategory 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 -like categories such as the category of cycles, Segal’s category , the tree category , 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. A further generalization which allows noninvertible level morphisms is a
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 and .
If is a direct category, then for any model category the colimit functor 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