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 $R$. It is named after Christopher Reedy.

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.

Definition

A Reedy category is a category $R$ equipped with two wide subcategories $R_+$ and $R_-$ and a total ordering on the objects of $R$, defined by a degree function $d:Ob(R) \to \alpha$, 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 $R_+$. (Thus $(R_-,R_+)$ is a strict factorization system.)

Examples

The simplex category

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.)

Definition

(Reedy structure on the simplex category)
The Reedy category structure on $\Delta$ is defined by:

• The degree function $d \colon Ob(\Delta) \to \mathbb{N}$ is defined by $[k] \mapsto k$.

• 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.

And the Reedy category structure on $\Delta^{op}$ is defined by switching $\Delta_+$ and $\Delta_-$.

Further examples

• 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: use the same degree function, and exchange $R_+$ and $R_-$.

• The integers regarded as a poset is NOT a Reedy category, since it is not well-founded in either direction.

• 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.

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.

Dually, so is the subcategory $R_-$ of any Reedy category considered as a category in its own right. This amounts to “discarding the faces” in a shape category.

Generalized Reedy categories

One problem with the notion of Reedy category is that it violates the principle of equivalence: 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!

Proof: Take any isomorphism $f$, let $f = g h$ and $h f^{-1} = g' h'$ be the unique factorizations. Then $id = g h f^{-1} = (g g') h'$, so $h' = id$ and $g g' = id$, whence $g = id$ and $g' = id$ since $g, g' \in R_+$. Thus $f = h \in R_-$. The same argument applied to $f^{-1}$ shows that $f$ preserves the degree, hence $f = id$. $\qed$

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. There is a similar notion (which however does not comprise all Reedy categories) due to Denis-Charles Cisinski. A further generalization which allows noninvertible level morphisms is a

• c-Reedy category.

Elegant Reedy categories

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 $\Theta$.

Eilenberg-Zilber categories

Eilenberg-Zilber categories are a special sort of generalized Reedy category that behave rather like an elegant strict Reedy category.

Enriched Reedy categories

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

Reedy categories with fibrant constants.

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.

References

See the references at Reedy model structure