nLab
schematic homotopy type

Context

Higher geometry

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

A schematic homotopy type is a geometric ∞-stack over a site of formal duals of kk-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.

(…)

Definition

Let kk be a commutative ring, TT the Lawvere theory of commutative kk-associative algebras. Let 𝕌𝕍\mathbb{U} \subset \mathbb{V} be an inclusion of universes Let

TC=TAlg 𝕌TAlg 𝕍 T \hookrightarrow C = T Alg_{\mathbb{U}} \hookrightarrow T Alg_{\mathbb{V}}

be the site on formal duals of small kk-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

(𝒪Spec):(TAlg 𝕍 Δ) opSpec𝒪Sh (C)=:H (\mathcal{O} \dashv Spec) : (T Alg_{\mathbb{V}}^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh_\infty(C) =: \mathbf{H}

(due to Toën) where the (∞,1)-topos H\mathbf{H} is the (∞,1)-category of (∞,1)-sheaves on CC.

Definition

(…) Let PerfHPerf \in \mathbf{H} be the stack of perfect complexes of modules on CC. (…)

Write PMor(H)P \subset Mor(\mathbf{H}) for the class of morphisms such that for all pPp \in P we have that H(p,Perf)\mathbf{H}(p,Perf) is an equivalence.

This is discussed in (HirschowitzSimpson, paragraph 21).

Definition

A pointed schematic homtopy type is the delooping BGH\mathbf{B}G \in \mathbf{H} of an ∞-group GHG \in \mathbf{H} such that

  • GG is in the image of SpecSpec, in that there is ATAlg ΔA \in T Alg^\Delta such that GSpecAG \simeq Spec A;

  • BG\mathbf{B}G is a PP-local object.

This appears as (Toën, def 3.1.2)

Properties

Observation

A schematic homotopy type is in particular a geometric ∞-stack over CC.

Examples

de Rham schematic homotopy type

For a connected scheme XX let X dRX_{dR} be its de Rham space. According to Toën, sect. 3.5.1 one finds that the functor

Ho(SchHoType/)Set Ho(SchHoType/\mathbb{C}) \to Set
FHo Sh (,1)(Alg op)(X dR,F) F \mapsto Ho_{Sh_{(\infty,1)}(Alg_\mathbb{C}^{op})}(X_{dR}, F)

is co-representable by a schematic homotopy type X dRX^{dR}. This is the de Rham schematic homotopy type. The cohomology of X dRSh (,1)X^{dR} \in Sh_{(\infty,1)} is the algebraic de Rham cohomology of XX.

Remark

A similar construction exists in every cohesive (∞,1)-topos. See the discussion in the section cohesive (∞,1)-topos – de Rham cohomology.

References

An introduction to the general theory

The stack PerfPerf 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.