(∞,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 schematic homotopy type is a geometric ∞-stack over a site of formal duals of -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 be a commutative ring, the Lawvere theory of commutative -associative algebras. Let be an inclusion of universes Let
be the site on formal duals of small -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 is the (∞,1)-category of (∞,1)-sheaves on .
(…) Let be the stack of perfect complexes of modules on . (…)
Write for the class of morphisms such that for all we have that is an equivalence.
This is discussed in (HirschowitzSimpson, paragraph 21).
A pointed schematic homtopy type is the delooping of an ∞-group such that
is in the image of , in that there is such that ;
is a -local object.
This appears as (Toën, def 3.1.2)
de Rham schematic homotopy type
For a connected scheme let 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 . This is the de Rham schematic homotopy type. The cohomology of is the algebraic de Rham cohomology of .
An introduction to the general theory
The stack of perfect complexes is discussed for instance in section 21 of