higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
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 -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)
A schematic homotopy type is in particular a geometric ∞-stack over .
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 .
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 of perfect complexes is discussed for instance in section 21 of
Last revised on November 9, 2014 at 07:39:00. See the history of this page for a list of all contributions to it.