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 schematic homotopy type is a geometric ∞-stack over a site of formal duals of $k$-algebras that models a homotopy type in generalization to how a dg-algebra models a rational space in rational homotopy theory: schematic homotopy type can in particular model more general fundamental groups.
(…)
Let $k$ be a commutative ring, $T$ the Lawvere theory of commutative $k$-associative algebras. Let $\mathbb{U} \subset \mathbb{V}$ be an inclusion of universes Let
be the site on formal duals of small $k$-algebras equipped with the fpqc-topology.
By the general discussion at function algebras on ∞-stacks we have then the Isbell duality pair of adjoint (∞,1)-functors
(due to Toën) where the (∞,1)-topos $\mathbf{H}$ is the (∞,1)-category of (∞,1)-sheaves on $C$.
(…) Let $Perf \in \mathbf{H}$ be the stack of perfect complexes of modules on $C$. (…)
Write $P \subset Mor(\mathbf{H})$ for the class of morphisms such that for all $p \in P$ we have that $\mathbf{H}(p,Perf)$ is an equivalence.
This is discussed in (HirschowitzSimpson, paragraph 21).
A pointed schematic homtopy type is the delooping $\mathbf{B}G \in \mathbf{H}$ of an ∞-group $G \in \mathbf{H}$ such that
$G$ is in the image of $Spec$, in that there is $A \in T Alg^\Delta$ such that $G \simeq Spec A$;
$\mathbf{B}G$ is a $P$-local object.
This appears as (Toën, def 3.1.2)
A schematic homotopy type is in particular a geometric ∞-stack over $C$.
For a connected scheme $X$ let $X_{dR}$ be its de Rham space. According to Toën, sect. 3.5.1 one finds that the functor
is co-representable by a schematic homotopy type $X^{dR}$. This is the de Rham schematic homotopy type. The cohomology of $X^{dR} \in Sh_{(\infty,1)}$ is the algebraic de Rham cohomology of $X$.
A similar construction exists in every cohesive (∞,1)-topos. See the discussion in the section cohesive (∞,1)-topos – de Rham cohomology.
An introduction to the general theory
Ludmil Katzarkov, Tony Pantev, Bertrand Toën, Schematic homotopy types and non-abelian Hodge theory, math.AG/0107129
Bertrand Toën, Affine stacks (Champs affines) (arXiv:math/0012219)
The stack $Perf$ of perfect complexes is discussed for instance in section 21 of