nLab homotopical algebraic geometry

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

• the ordinary category of sets (yielding ordinary algebraic geometry)
• the (infinity,1)-category of simplicial commutative rings (yielding derived algebraic geometry),
• the (infinity,1)-category of spectra (yielding spectral algebraic geometry).

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.

Related concepts

• derived algebraic geometry
• spectral algebraic geometry

References

• Jacob Lurie, Derived Algebraic Geometry, thesis, pdf.

• Jacob Lurie, Structured Spaces.

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

• Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry I: topos theory, 2002, arXiv:math/0207028.

• Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry II: geometric stacks and applications, 2004, arXiv:math/0404373.

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

• Mauro Porta, Tony Yue Yu, Higher analytic stacks and GAGA theorems, arXiv:1412.5166.

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.

• Scott Balchin, Augmented Homotopical Algebraic Geometry, (arXiv:1711.02640)