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Reedy category with fibrant constants

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Context

Model category theory

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Homotopy theory

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

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see also algebraic topology

Introductions

Definitions

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Homotopy groups

Basic facts

Theorems

Contents

Definition

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.

Properties

Theorem

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.

Proposition
  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.
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=ΔR = \Delta, but the argument given there applies to any elegant Reedy category.

Examples

  • 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.

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.