higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A geometric $\infty$-stack is an ∞-stack over a geometry with function theory $(\mathcal{O} \dashv Spec)$ which is an ∞-groupoid that is degreewise an $\infty$-stack in the image of $Spec$.
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 $T$ via Isbell duality. Write $T Alg$ for the category of algebras over a Lawvere theory and write $T Alg^{\Delta}$ for the (∞,1)-category of cosimplicial $T$-algebras .
Consider a site $T \subset C \subset T Alg^{op}$ 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 $\mathbf{H} := Sh_\infty(C)$ is the (∞,1)-category of (∞,1)-sheaves over $C$, the big topos for the higher geometry over $C$.
An object $X \in \mathbf{H}$ is called a geometric $\infty$-stack over $C$ if there is it is the (∞,1)-colimit
over a groupoid object $K_\bullet : \Delta \to \mathbf{H}$ in $\mathbf{H}$ such that
$K_0$ and $K_1$ are in the image of $Spec : (T Alg^{\Delta})^{op} \to \mathbf{H}$;
the target map $d_1 : K_1 \to K_0$ is (…sufficiently well behaved…)
For $T$ the theory of commutative associative algebras over a commutative ring $k$ and $C$ the fpqc topology this appears as Toën 00, definition 4.1.4.
Geometric $\infty$-stacks are stable under (∞,1)-pullbacks along morphism in the image of $Spec$.
Use that in the (∞,1)-topos $\mathbf{H}$ we have universal colimits and that $Spec$ is right adjoint.
A presentation of geometric $\infty$-stacks, in some generality, by suitably Kan-fibrant simplicial objects is in (Pridham 09). See also at Kan-fibrant simplicial manifold.
Every object in the image of $Spec : T Alg_\infty^{op} \to \mathbf{H}$ is a geometric $\infty$-stack.
Over the étale site an algebraic stack that is a geometric stack is also a geometric $\infty$-stack.
Every schematic homotopy type is given by a geometric $\infty$-stack.
Kan-fibrant simplicial manifolds serve to present geometric $\infty$-stacks in higher differential geometry.
geometric $\infty$-stack, function algebras on ∞-stacks
The notion of geometric $\infty$-stack as a weak quotient of affine $\infty$-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
Jonathan Pridham, Representability of derived stacks (arXiv:1011.2189)
Discussion of presentation of geometric $\infty$-stacks by Kan-fibrant simplicial objects in the site is in