nLab
locally representable structured (infinity,1)-topos

Idea

For $𝒢$ a geometry (for structured (∞,1)-toposes) a $𝒢$-structured (∞,1)-topos $(𝒳,{𝒪}_{𝒳})$ is locally representable if it is locally equivalent to $\mathrm{Spec}U$ for $U\in \mathrm{Pro}(𝒢)$ (the pro-objects in an (∞,1)-category), or $U\in 𝒢$ 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 $𝒢$ be a geometry (for structured (∞,1)-toposes). Write ${𝒢}_0$ for the underlying discrete geometry. The identity functor

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

is then a morphism of geometries.

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

Affine $𝒢$-schemes

Theorem ( StSp 2.1.1 )

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

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

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

Remark ( StSp p. 38 )

There is a canonical morphism

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

Definition ( affine $𝒢$-scheme, StSp 2.3.9)

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

$${\mathrm{Spec}}^{𝒢}:\mathrm{Pro}(𝒢{)}^{\mathrm{op}}\stackrel{\mathrm{can}}{\to }\mathrm{LTop}({𝒢}_0)\stackrel{{\mathrm{Spec}}_{{𝒢}_0}^{𝒢}}{\to }\mathrm{LTop}(𝒢).$$\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 $𝒢$-structured (∞,1)-topos in the image of this functor is an affine $𝒢$-scheme.

$𝒢$-Schemes

Definition (geometric scheme, StSp 2.3.9)

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

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

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

such that

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

• for every $U_i$ there exists an equivalence

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

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

Definition (pregeometric scheme, StSp, 3.4.6)

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

This means that for $f:𝒯\to 𝒢$ the geometric envelope and for $𝒪{\prime }_{𝒳}$ the $𝒢$-structure on $𝒳$ such that ${𝒪}_{𝒳}\simeq 𝒪{\prime }_{𝒳}\circ f$, we have that $(𝒳,𝒪{\prime }_{𝒳})$ is a $𝒢$-scheme.

Smooth $𝒢$-schemes

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

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

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

Definition ( smooth $𝒢$-scheme, StSp 3.5.6)

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

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 ${𝒯}_{\mathrm{Zar}}(k)$.

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

Derived Deligne-Mumford stacks

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

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

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

Derived schemes with $E_{\mathrm{\infty }}$-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

Derived smooth manifolds

• derived smooth manifold.

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 $𝒢$-schemes (absolute spectra) is in section 2.2.