# nLab simplicial resolution

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

In any ambient category or $\infty$-category $C$ that admits a notion of colimit or weak colimit, a simplicial resolution of an object $c \in C$ is a simplicial object $y_\bullet : \Delta^{op} \to C$ such that it realizes $c$ as a colimit

$c \simeq colim_{[k]} y_k \,.$

The term is also used for a Reedy fibrant replacement of a constant simplicial object in a model category; see also resolution.

## Examples

• In an (infinity,1)-topos Čech cover $C(U) \stackrel{\simeq}{\to} X$ induced by a cover $(U = \coprod_i U_i) \to X$ is a simplicial resolution of $X$.

## References

Simplicial resolutions in the context of presentable (infinity,1)-categories are discussed in section 6.1.4 of

(below lemma 6.1.4.3)

Revised on March 11, 2011 20:27:42 by Mike Shulman (128.54.63.224)