nLab
homotopical algebraic geometry

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 CC, let CAlg(C)CAlg(C) denote the (infinity,1)-category of commutative algebra objects in CC.

Definition

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

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

Definition

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

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

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

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

Definition

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

αX αX \coprod_\alpha X_\alpha \longrightarrow X

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

Last revised on April 4, 2015 at 20:04:00. See the history of this page for a list of all contributions to it.