nLab sifted (infinity,1)-colimit

Definition

A sifted $(\infty,1)$-colimit is an (∞,1)-colimit over a diagram that is a sifted (∞,1)-category.

Properties

Proposition

Let $C$ be an (∞,1)-category such that products preserve sifted (∞,1)-colimits (for instance an (∞,1)-topos, see universal colimits).

Then sifted (∞,1)-colimits preserve finite homotopy products.

(Lurie HTT, lemma 5.5.8.11).

Examples

Simplicial $\infty$-colimits

Proposition

(simplicial $\infty$-colimits are sifted)
The $(\infty,1)$-colimits of simplicial objects in an $(\infty,1)$-category are sifted.

This is because the opposite of the simplex category is a sifted (∞,1)-category (Lurie HTT, Prop. 5.5.8.4).

Remark

Simplicial $\infty$-colimits preserve even homotopy fiber products, under mild conditions: see at geometric realization of simplicial topological spaces the section Preservation of homotopy limits.

Related concepts

• sifted colimit, sifted $(\infty,1)$-colimit

• filtered colimit, filtered (∞,1)-colimit

References

• Jacob Lurie, Higher Topos Theory