# nLab Reedy category with fibrant constants

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

# Contents

## 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

Last revised on October 7, 2013 at 09:05:26. See the history of this page for a list of all contributions to it.