nLab Reedy category with fibrant constants



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




A Reedy category with fibrant constants is a Reedy category RR such that for every model category MM and every fibrant object XMX \in M the constant RR-diagram at XX is Reedy fibrant.



For a Reedy category RR the following are equivalent.

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

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

  1. The product of two Reedy categories with fibrant constants is a Reedy category with fibrant constants.
  2. If RR is an elegant Reedy category and XX is a presheaf on RR, then the category of elements of XX is a Reedy category with fibrant constants.
  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=ΔR = \Delta, but the argument given there applies to any elegant Reedy category.


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

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

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


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