homotopical algebraic geometry




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.


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


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}.


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).


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.


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 SpcSpc to those valued in augmentation categories, a special class of generalized Reedy categories, see.

Last revised on January 16, 2021 at 10:43:30. See the history of this page for a list of all contributions to it.