(∞,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
The concept of geometric -stack is the refinement to ∞-stack of that of geometric stack.
There is an intrinsic definition which iterates that of geometric stacks and says inductively that a geometric -stack is one which has an -atlas and such that its diagonal is -representable (Toën-Vezzosi 04, def. 188.8.131.52).
Then there is a result which says that such geometric -stacks are equivalently those represented by suitable Kan complex-objects in the given site (“internal infinity-groupoids” in the site) (Pridham 09).
(There is also a definition of “geometric -stack” in (Toën 00, definition 4.1.4), which is however different.)
Presentation by Kan-fibrant simplicial objects
A presentation of geometric -stacks, in some generality, by suitably Kan-fibrant simplicial objects is in (Pridham 09). See also at Kan-fibrant simplicial manifold.
The text below follows (Toën 00). Needs to be connected to the rest of the entry.
We consider the higher geometry encoded by a Lawvere theory via Isbell duality. Write for the category of algebras over a Lawvere theory and write for the (∞,1)-category of cosimplicial -algebras .
Consider a site that satisfies the assumptions described at function algebras on ∞-stacks. Then, by the discussion given there, we have a pair of adjoint (∞,1)-functors
where is the (∞,1)-category of (∞,1)-sheaves over , the big topos for the higher geometry over .
An object is called a geometric -stack over if there is it is the (∞,1)-colimit
over a groupoid object in such that
and are in the image of ;
the target map is lisse.
For the theory of commutative associative algebras over a commutative ring and the fpqc topology this appears as (Toën 00, definition 4.1.4).
Geometric -stacks are stable under (∞,1)-pullbacks along morphism in the image of .
The notion of geometric -stack as a weak quotient of affine -stacks is considered in section 4 of
More general theory in the context of derived algebraic geometry is in
and specifically in E-∞ geometry in
Discussion of presentation of geometric -stacks by Kan-fibrant simplicial objects in the site is in