nLab
relative (infinity,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^{\triangleright} \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)

Examples

  • If 𝒟=Δ[0]\mathcal{D} = \Delta[0] is the terminal category and f:𝒞Δ[0]f: \mathcal{C} \to \Delta[0] is the unique functor, then an ff-colimit is the same thing as a colimit in the usual sense.

  • If K=Δ[0]K = \Delta[0] is the terminal category so that p:Δ[0]𝒞p: \Delta[0] \to \mathcal{C} picks out an object and fp¯:K =Δ[1]𝒟f\overline{p}: K^{\triangleright} = \Delta[1] \to \mathcal{D} picks out an edge, an ff-colimit is precisely an ff-cocartesian lift.

  • If f:𝒞𝒟f: \mathcal{C} \to \mathcal{D} is a cocartesian fibration such that the fibers 𝒞 s\mathcal{C}_s have all KK-shaped colimits and the reindexing functors 𝒞 s𝒞 s\mathcal{C}_s \to \mathcal{C}_{s'} preserve KK-shaped colimits for each morphism sss \to s' in 𝒟\mathcal{D}, then for any extension g:K 𝒟g: K^\triangleright \to \mathcal{D} of fpfp, the relative colimit is given by the functor p¯:K 𝒞\bar{p}: K^\triangleright \to \mathcal{C} which extends pp to the fiber 𝒞 g()\mathcal{C}_{g(\infty)} (where K \infty \in K^\triangleright is the cocone point) in a cocartesian way, and carries the cocone point of K K^\triangleright to the colimit of the induced diagram K𝒞 g()K \to \mathcal{C}_{g(\infty)}. (See Lurie, Cor 4.3.1.11 and its proof.)

References

Last revised on February 27, 2021 at 10:22:38. See the history of this page for a list of all contributions to it.