nLab infinity-stackification

Context

$(\infty,1)$-Topos theory

(∞,1)-topos theory

Constructions

structures in a cohesive (∞,1)-topos

Contents

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.

References

For instance section 6.5.3 of

Revised on April 30, 2012 13:59:52 by Urs Schreiber (89.204.137.196)