nLab
simplicial resolution

Context

Higher category theory

higher category theory

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Contents

Idea

In any ambient category or \infty-category CC that admits a notion of colimit or weak colimit, a simplicial resolution of an object cCc \in C is a simplicial object y :Δ opCy_\bullet : \Delta^{op} \to C such that it realizes cc as a colimit

ccolim [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)XC(U) \stackrel{\simeq}{\to} X induced by a cover (U= iU i)X(U = \coprod_i U_i) \to X is a simplicial resolution of XX.

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)