nLab relative (infinity,1)-limit

Idea

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

Definition

Definition

For $f \colon \mathcal{C} \to \mathcal{D}$ 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

$$\overline{p} \colon K^{\triangleright} \to \mathcal{C}$$

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

$$p \coloneqq \overline{p}|K \,,$$

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

$$\mathcal{C}_{\overline{p}/} \to \mathcal{C}_{p/} \underset{\mathcal{D}_{f p /}}{\times} \mathcal{D}_{f\overline{p}/}$$

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

(Lurie, def. 4.3.1.1)

Examples

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

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

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

References

• Jacob Lurie, Higher Topos Theory