nLab locally representable structured (infinity,1)-topos

Idea

For $\mathcal{G}$ a geometry (for structured (∞,1)-toposes) a $\mathcal{G}$-structured (∞,1)-topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is locally representable if it is locally equivalent to $Spec U$ for $U \in Pro(\mathcal{G})$ (the pro-objects in an (∞,1)-category), or $U \in \mathcal{G}$ itself if it is locally finite presented .

This generalizes

• the notion of smooth manifold from differential geometry;

• the notion of scheme from algebraic geometry.

• etc.

Definition

Let $\mathcal{G}$ be a geometry (for structured (∞,1)-toposes). Write $\mathcal{G}_0$ for the underlying discrete geometry. The identity functor

$$p : \mathcal{G}_0 \to \mathcal{G}$$

is then a morphism of geometries.

Recall the notation $LTop(\mathcal{G})$ for the (∞,1)-category of $\mathcal{G}$-structured (∞,1)-toposes and geometric morphisms between them.

Affine $\mathcal{G}$-schemes

Theorem ( StSp 2.1.1 )

There is a pair of adjoint (∞,1)-functors

$$p^* : LTop(\mathcal{G}) \stackrel{\leftarrow}{\to} LTop(\mathcal{G}_0) : \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}$$

with $\mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}$ left adjoint to the canonical functor $p^*$ given by precomposition with $p$.

Remark ( StSp p. 38 )

There is a canonical morphism

$$can : Pro(\mathcal{G})^{op} \to LTop(\mathcal{G}_0)$$

Definition ( affine $\mathcal{G}$-scheme, StSp 2.3.9)

Write $\mathbf{Spec}^{\mathcal{G}}$ for the (∞,1)-functor

$$\mathbf{Spec}^{\mathcal{G}} : Pro(\mathcal{G})^{op} \stackrel{can}{\to} LTop(\mathcal{G}_0) \stackrel{ \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}} }{\to} LTop(\mathcal{G}) \,.$$

A $\mathcal{G}$-structured (∞,1)-topos in the image of this functor is an affine $\mathcal{G}$-scheme.

$\mathcal{G}$-Schemes

Definition (geometric scheme, StSp 2.3.9)

Let $\mathcal{G}$ be a geometry (for structured (∞,1)-toposes).

A $\mathcal{G}$-structured (∞,1)-topos $(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ is a $\mathcal{G}$-scheme if

• there exists a collection $\{U_i \in \mathcal{X}\}$

such that

• the $\{U_i\}$ cover $\mathcal{X}$ in that the canonical morphism $\coprod_i U_i \to {*}$ (with ${*}$ the terminal object of $\mathcal{X}$) is an effective epimorphism;

• for every $U_i$ there exists an equivalence

  $$(\mathcal{X}/{U_i}, \mathcal{O}_{\mathcal{X}}|_{U_i}) \simeq \mathbf{Spec}^{\mathcal{G}} A_i$$

  of structured $(\infty,1)$-toposes for some $A_i \in Pro(\mathcal{G})$ (in the (∞,1)-category of pro-objects of $\mathcal{G}$).

Definition (pregeometric scheme, StSp, 3.4.6)

For $\mathcal{T}$ a pregeometry, a $\mathcal{T}$-structured (infinity,1)-topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a $\mathcal{T}$-scheme if it is a $\mathcal{G}$-scheme for the geometric envelope $\mathcal{G}$ of $\mathcal{T}$.

This means that for $f : \mathcal{T} \to \mathcal{G}$ the geometric envelope and for $\mathcal{O}'_{\mathcal{X}}$ the $\mathcal{G}$-structure on $\mathcal{X}$ such that $\mathcal{O}_{\mathcal{X}} \simeq \mathcal{O}'_{\mathcal{X}} \circ f$, we have that $(\mathcal{X}, \mathcal{O}'_{\mathcal{X}})$ is a $\mathcal{G}$-scheme.

Smooth $\mathcal{G}$-schemes

Let $\Tau$ be a pregeometry (for structured (∞,1)-toposes) and let $\Tau \hookrightarrow \mathcal{G}$ be an inclusion into an enveloping geometry (for structured (∞,1)-toposes).

We think of the objects of $\Tau$ as the smooth test spaces – for instance the cartesian products of some affine line $R$ with itsef – and of the objects of $\mathcal{G}$ as affine test spaces that may have singular points where they are not smooth.

The idea is that a smooth $\mathcal{G}$-scheme is a $\mathcal{G}$-structured space that is locally not only equivalent to objects in $\mathcal{G}$, but even to the very nice – “smooth” – objects in $\mathcal{Tau}$.

Definition ( smooth $\mathcal{G}$-scheme, StSp 3.5.6)

With an envelope $\Tau \hookrightarrow \mathcal{G}$ fixed, a $\mathcal{G}$-scheme is called smooth if there the affine schemes $\mathbf{Spec}^{\mathcal{G}} A_i$ appearing in its definition may be chosen with $A_i$ in the image of the includion $\tau \hookrightarrow \mathcal{G}$.

Examples

Ordinary schemes

See the discussion at derived scheme for how ordinary schemes are special cases of generalized schemes.

Ordinary Deligne-Mumford stacks

See the discussion at derived Deligne-Mumford stack for how ordinary Deligne-Mumford stacks are special cases of derived Deligne-Mumford stacks.

Derived schemes

Definition (derived scheme, Structured Spaces, 4.2.8)

Let $k$ be a commutative ring. Recall the pregoemtry $\mathcal{T}_{Zar}(k)$.

A derived scheme over $k$ is a $\mathcal{T}_{Zar}(k)$-scheme.

Derived smooth manifolds

• derived smooth manifold.

Derived Deligne-Mumford stacks

Definition (derived Deligne-Mumford stack, Structured Spaces, 4.3.19)

Let $k$ be a commutative ring. Recall the pregeometry $\mathcal{T}_{et}(k)$

A derived Deligne-Mumford stack over $k$ is a $\mathcal{T}_{et}(k)$-scheme.

Derived schemes with $E_\infty$-ring valued structure sheaves

The above derived schemes have structure sheaves with values in simplicial commutative rings. There is also a notion of derived scheme whose structure sheaf takes values in E-infinity rings. The theory of these is to be described in full detail in

• Jacob Lurie, Spectral Schemes .

An indication of some details is in

• Paul Goerss, Topological Algebraic Geometry - A Workshop

See at E-∞ scheme and E-∞ geometry.

Related concepts

• geometry (for structured (∞,1)-toposes)

• structured (∞,1)-topos

• locally representable structured (∞,1)-topos

References

Generalized schemes are definition 2.3.9 of

• Jacob Lurie, Derived Algebraic Geometry V - Structured Spaces

The definition of affine $\mathcal{G}$-schemes (absolute spectra) is in section 2.2.