(∞,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 geometric -stack is an ∞-stack over a geometry with function theory which is an ∞-groupoid that is degreewise an -stack in the image of .
This generalizes the notion of geometric stack from topos theory to (∞,1)-topos theory.
The text below follows (Toën 00). Needs to be expanded and (Toën-Vezzosi 04, Lurie) needs to be brought into the picture…
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 (…sufficiently well behaved…)
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 .
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 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