(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
A (Grothendieck) -topos is the (n,1)-category version of a Grothendieck topos: a collection of (n-1)-groupoid-valued sheaves on an -categorical site.
Notice that an ∞-stack on an ordinary (1-categorical) site that takes values in ∞-groupoids which happen to by 0-truncated, i.e. which happen to take values just in Set ∞Grpd is the same as an ordinary sheaf of sets.
This generalizes: every -topos arises as the full (∞,1)-subcategory on -truncated objects in an (∞,1)-topos of -stacks on an (n,1)-category site.
a 1-Grothendieck topos is precisely an accessible geometric embedding into a category of presheaves on some small category
a (∞,1)-topos (of ∞-stacks/(∞,1)-sheaves) is precisely an accessible geometric embedding into a (∞,1)-category of (∞,1)-presheaves on some small (∞,1)-category :
This appears as HTT, def. 188.8.131.52.
Write (∞,1)-Topos for the (∞,1)-category of (∞,1)-topos and (∞,1)-geometric morphisms. Write for the (n+1,1)-category of -toposes and geometric morphisms between these.
The following proposition asserts that when passing to the -topos of an (∞,1)-topos , only the n-localic “Postnikov stage” of matters.
This is (HTT, prop. 184.108.40.206).
For any , -truncation induces a localization
that identifies equivalently with the full subcategory of -localic -toposes.
(This is 220.127.116.11 in view of the following remarks.)
If is a (2,1)-topos in which every object is covered by a 0-truncated object, then is equivalent to the category of (2,1)-sheaves on a 1-site (rather than merely a (2,1)-site, as is the case for general (2,1)-topoi), and is thus canonically associated to a 1-topos, namely the category of 1-sheaves on that same 1-site. And in fact, can be recovered from this 1-topos as the category of (2,1)-sheaves for its canonical topology.
See truncated 2-topos for more.
Section 6.4 of