nLab Reedy category with fibrant constants

Definition

A Reedy category with fibrant constants is a Reedy category $R$ such that for every model category $M$ and every fibrant object $X \in M$ the constant $R$-diagram at $X$ is Reedy fibrant.

Properties

Theorem

For a Reedy category $R$ the following are equivalent.

1. $R$ has fibrant constants.
2. For any model category $M$ the colimit functor $\colim_R \colon M^R \to M$ is a left Quillen functor.
3. All matching categories of $R$ are connected or empty.

This theorem is proven in Hirschhorn,Proposition 15.10.2 and Theorem 15.10.8.

Proposition

1. The product of two Reedy categories with fibrant constants is a Reedy category with fibrant constants.
2. If $R$ is an elegant Reedy category and $X$ is a presheaf on $R$, then the category of elements of $X$ is a Reedy category with fibrant constants.

Proof

1. This follows by combining point 3. of the theorem above with the fact that the matching categories satisfy the “Leibniz rule”, see Riehl, Verity, Observation 4.2.
2. This is proven in Hirschhorn, Proposition 15.10.4 in the case of $R = \Delta$, but the argument given there applies to any elegant Reedy category.

Examples

• The simplex category $\Delta$ is a Reedy category with fibrant constants.

• More generally, Joyal’s disk categories $\Theta_n$ are Reedy categories with fibrant constants.

• Every direct category is a Reedy category with fibrant constants.

References

• Philip Hirschhorn, Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs, American Mathematical Society, 2009,

• Emily Riehl, Dominic Verity The theory and practice of Reedy categories arXiv