# nLab sifted (infinity,1)-colimit

Contents

### Context

#### Limits and colimits

limits and colimits

# Contents

## 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.

## 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.

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

## 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.