# nLab schematic homotopy type

## Theorems

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

(∞,1)-topos theory

## Constructions

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: 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

$T↪C=T{\mathrm{Alg}}_{𝕌}↪T{\mathrm{Alg}}_{𝕍}$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

$\left(𝒪⊣\mathrm{Spec}\right):\left(T{\mathrm{Alg}}_{𝕍}^{\Delta }{\right)}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{←}}{\underset{\mathrm{Spec}}{\to }}{\mathrm{Sh}}_{\infty }\left(C\right)=:H$(\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 $P\subset \mathrm{Mor}\left(H\right)$ for the class of morphisms such that for all $p\in P$ we have that $H\left(p,\mathrm{Perf}\right)$ 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

$\mathrm{Ho}\left(\mathrm{SchHoType}/ℂ\right)\to \mathrm{Set}$Ho(SchHoType/\mathbb{C}) \to Set
$F↦{\mathrm{Ho}}_{{\mathrm{Sh}}_{\left(\infty ,1\right)}\left({\mathrm{Alg}}_{ℂ}^{\mathrm{op}}\right)}\left({X}_{\mathrm{dR}},F\right)$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 ${X}^{\mathrm{dR}}\in {\mathrm{Sh}}_{\left(\infty ,1\right)}$ 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

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

Revised on January 11, 2011 14:01:05 by Urs Schreiber (89.204.137.68)