nLab
simplicial resolution

Idea

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

$$c\simeq {\mathrm{colim}}_{[k]}{y}_k.$$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

• Jacob Lurie, Higher Topos Theory

(below lemma 6.1.4.3)