nLab
sifted (infinity,1)-colimit

Contents

Contents

Definition

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

Properties

Proposition

Let CC 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 (,1)(\infty,1)-colimits of simplicial objects in an ( , 1 ) (\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.

References

Last revised on September 21, 2021 at 06:24:54. See the history of this page for a list of all contributions to it.