nLab
schematic homotopy type

Idea

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.

(…)

Definition

Let $k$ be a commutative ring, $T$ the Lawvere theory of commutative $k$-associative algebras. Let $𝕌\subset 𝕍$ be an inclusion of universes Let

$$T\hookrightarrow C=T{\mathrm{Alg}}_{𝕌}\hookrightarrow T{\mathrm{Alg}}_{𝕍}$$T \hookrightarrow C = T Alg_{\mathbb{U}} \hookrightarrow T Alg_{\mathbb{V}}

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

$$(𝒪⊣\mathrm{Spec}):(T{\mathrm{Alg}}_{𝕍}^{\Delta }{)}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{\leftarrow }}{\underset{\mathrm{Spec}}{\to }}{\mathrm{Sh}}_{\mathrm{\infty }}(C)=:H$$(\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 $H$ is the (∞,1)-category of (∞,1)-sheaves on $C$.

Definition

(…) Let $\mathrm{Perf}\in H$ be the stack of perfect complexes of modules on $C$. (…)

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

This is discussed in (HirschowitzSimpson, paragraph 21).

Definition

A pointed schematic homtopy type is the delooping $BG\in H$ of an ∞-group $G\in H$ such that

• $G$ is in the image of $\mathrm{Spec}$, in that there is $A\in T{\mathrm{Alg}}^{\Delta }$ such that $G\simeq \mathrm{Spec}A$;

• $BG$ is a $P$-local object.

This appears as (Toën, def 3.1.2)

Properties

Observation

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

Examples

de Rham schematic homotopy type

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

$$\mathrm{Ho}(\mathrm{SchHoType}/ℂ)\to \mathrm{Set}$$Ho(SchHoType/\mathbb{C}) \to Set

$$F\mapsto {\mathrm{Ho}}_{{\mathrm{Sh}}_{(\mathrm{\infty },1)}({\mathrm{Alg}}_{ℂ}^{\mathrm{op}})}(X_{\mathrm{dR}},F)$$F \mapsto Ho_{Sh_{(\infty,1)}(Alg_\mathbb{C}^{op})}(X_{dR}, F)

is co-representable by a schematic homotopy type $X^{\mathrm{dR}}$. This is the de Rham schematic homotopy type. The cohomology of $X^{\mathrm{dR}}\in {\mathrm{Sh}}_{(\mathrm{\infty },1)}$ is the algebraic de Rham cohomology of $X$.

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 theor

• Ludmil Katzarkov, Tony Pantev, Bertrand Toën], Schematic homotopy types and non-abelian Hodge theory (pdf)

• Bertrand Toën, Affine stacks (Champs affines) (arXiv:math/0012219)

The stack $\mathrm{Perf}$ of perfect complexes is discussed for instance in section 21 of

• Andre Hirschowicz, Carlos Simpson, Descente pour les n-champs (arXiv:math/9807049)