# nLab schematic homotopy type

Contents

### Context

Ingredients

Concepts

Constructions

Examples

Theorems

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

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos

# Contents

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

(…)

## Definition

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

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

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

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

(due to Toën 2006) where the (∞,1)-topos $\mathbf{H}$ is the (∞,1)-category of (∞,1)-sheaves on $C$.

###### Definition

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

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

This is discussed in (HirschowitzSimpson, paragraph 21).

###### Definition

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

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

• $\mathbf{B}G$ is a $P$-local object.

This appears as (Toën 2006, def 3.1.2)

## Properties

###### Proposition

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

## Examples

### de Rham schematic homotopy type

For a connected scheme $X$ let $X_{dR}$ be its de Rham space. According to Toën 2006, sect. 3.5.1 one finds that the functor

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

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

An introduction to the general theory

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