nLab schematic homotopy type



Higher geometry

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

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



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 (via the fundamental theorem of dg-algebraic rational homotopy theory): schematic homotopy types in particular model more general fundamental groups



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 2006) where the (∞,1)-topos H\mathbf{H} is the (∞,1)-category of (∞,1)-sheaves on CC.


(…) 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).


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 2006, def 3.1.2)



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


de Rham schematic homotopy type

For a connected scheme XX let X dRX_{dR} be its de Rham space. According to Toën 2006, 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.


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

The stack PerfPerf of perfect complexes is discussed for instance in section 21 of

Last revised on September 23, 2021 at 17:55:38. See the history of this page for a list of all contributions to it.