nLab
infinity-stackification

Idea

$\mathrm{\infty }$-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

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

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

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

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

Examples

• A concrete model for such $\mathrm{\infty }$-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 $\mathrm{\infty }$-stackification is given by fibrant replacement in the model category.