nLab
infinity-stackification

Context

(,1)(\infty,1)-Topos theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

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 SS a site, sheafification is the functor

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

which sends every presheaf FF on SS to another presheaf F¯\bar F which is weakly equivalent to FF with respect to the homotopical category structure on PSh(S)PSh(S) induced from the Grothendieck topology on XX. The presheaf F¯\bar F respects weak equivalences and satisfies descent in that the hom-functor Hom PSh(S)(,F¯)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 “(,1)(\infty,1)-sheafification”, and so on.

Examples

References

For instance section 6.5.3 of

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