Contents

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Idea

Homotopical algebraic geometry is a homotopical generalization of algebraic geometry, where the affine schemes are not necessarily commutative algebras in the usual sense, but rather commutative algebra objects in an arbitrary symmetric monoidal (infinity,1)-category. In other words, it is the specialization of higher geometry where the local models are commutative algebras in a symmetric monoidal (infinity,1)-category.

Interesting examples can be obtained by starting with

As in ordinary algebraic geometry, there are two equivalent definitions of scheme: the so-called functor of points approach, or the approach via locally ringed toposes; we consider the former approach below.

## Definition

Given a symmetric monoidal (infinity,1)-category $C$, let $CAlg(C)$ denote the (infinity,1)-category of commutative algebra objects in $C$.

###### Definition

A $C$-prestack is an infinity-prestack on the opposite (infinity,1)-category of $CAlg(C)$. In other words it is a functor $CAlg(C) \to Spc$ to the (infinity,1)-category of spaces.

Let $\tau$ be a subcanonical Grothendieck topology on $CAlg(C)^{op}$.

###### Definition

A $(C,\tau)$-stack is an infinity-stack on the opposite (infinity,1)-category of $CAlg(C)$, with respect to the topology $\tau$. Let $Stk(C, \tau)$ denote the (infinity,1)-category of $(C,\tau)$-stacks. The Yoneda embedding is denoted

$Spec : CAlg(C) \hookrightarrow Stk(C,\tau)$

and a stack is called affine if it is in the essential image.

Let $P$ be a class of morphisms in $Stk(C, \tau)$ which is stable by base change. The triple $(C, \tau, P)$ is a called a homotopical algebraic geometry context (HAG context).

###### Definition

A $(C,\tau, P)$-scheme is a $(C,\tau)$-stack $X$ such that there exists a family of morphisms $(X_\alpha \to X)_\alpha$, each in $P$, such that $X_\alpha$ are affine and the induced morphism

$\coprod_\alpha X_\alpha \longrightarrow X$

is an epimorphism. Let $Sch(C, \tau, P)$ denote the (infinity,1)-category of $(C,\tau,P)$-schemes.

## References

An alternate exposition of the theory, using the presentations by model categories (hence the various model structures on simplicial presheaves), is given in

An exposition in the style of Toën-Vezzosi but in the language of (infinity,1)-categories can be found in

For an extension of homotopical algebraic geometry from $\infty$-stacks valued in $Spc$ to those valued in augmentation categories, a special class of generalized Reedy categories, see.

Last revised on January 16, 2021 at 15:43:30. See the history of this page for a list of all contributions to it.