nLab infinity-stackification

Idea

$\infinity$-Stackification is another term for (∞,1)-sheafification. It is the direct (∞,1)-categorical analog of the following 1-categorical situation.

Recall that for $S$ a site, sheafification is the functor

$$\bar{(-)} : PSh(S) \to Sh(S) \hookrightarrow PSh(S)$$

which sends every presheaf $F$ on $S$ to another presheaf $\bar F$ which is weakly equivalent to $F$ with respect to the homotopical category structure on $PSh(S)$ induced from the Grothendieck topology on $X$. The presheaf $\bar F$ respects weak equivalences and satisfies descent in that the hom-functor $Hom_{PSh(S)}(-,\bar F)$ sends weak equivalences (the local isomorphisms) to weak equivalences.

Essentially by definition (according to Higher Topos Theory) the situation for $\infty$-stacks is entirely analogous, as described at (∞,1)-category of (∞,1)-sheaves.

(Noticing that “$\infty$-stack” is synonymous to “(∞,1)-sheaf”, “$\infty$-stackification” to “$(\infty,1)$-sheafification”, and so on.

Examples

• A concrete model for such $\infty$-stacks is the Brown-Joyal-Jardine-Toën model using a model structure on simplicial presheaves. With respect to this model category structure on simplicial presheaves the fibrant objects are the globally Kan complex-valued (i.e. ∞-groupoid-valued) presheaves that satisfy descent for simplicial presheaves. Therefore here $\infty$-stackification is given by fibrant replacement in the model category.

Related concepts

• sheafification, plus-construction on presheaves

• (∞,1)-sheafification / ∞-stackification

References

For instance section 6.5.3 of

• Jacob Lurie, Higher Topos Theory