(∞,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.
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, 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
Revised on November 4, 2011 13:53:33