nLab
relative (∞,1)-limit

Contents

Context

(,1)(\infty,1)-Category theory

Limits and colimits

Contents

Idea

The relative version of the notion of (∞,1)-limit.

Definition

Definition

For f:𝒞𝒟f \colon \mathcal{C} \to \mathcal{D} an (∞,1)-functor between (∞,1)-categories which is presented by an inner fibration of quasi-categories (which we denote by the same symbols), and for

p¯:K *𝒞 \overline{p} \colon K^{\ast} \to \mathcal{C}

a cocone diagram in 𝒞\mathcal{C} over the KK-shaped diagram

pp¯|K, p \coloneqq \overline{p}|K \,,

then p¯\overline{p} is an (,1)(\infty,1)-colimiting cocone if the canonical map

𝒞 p¯/𝒞 p/×𝒟 fp/𝒟 fp¯/ \mathcal{C}_{\overline{p}/} \to \mathcal{C}_{p/} \underset{\mathcal{D}_{f p /}}{\times} \mathcal{D}_{f\overline{p}/}

(from the co-slice quasi-category) is an acyclic Kan fibration of simplicial sets.

(Lurie, def. 4.3.1.1)

References

Created on June 11, 2013 at 12:00:34. See the history of this page for a list of all contributions to it.