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locally representable structured (infinity,1)-topos

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Higher geometry

Contents

Idea

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

This generalizes

Definition

Let 𝒢\mathcal{G} be a geometry (for structured (∞,1)-toposes). Write 𝒢 0\mathcal{G}_0 for the underlying discrete geometry. The identity functor

p:𝒢 0𝒢 p : \mathcal{G}_0 \to \mathcal{G}

is then a morphism of geometries.

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

Affine 𝒢\mathcal{G}-schemes

Theorem ( StSp 2.1.1 )

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

p *:LTop(𝒢)LTop(𝒢 0):Spec 𝒢 0 𝒢 p^* : LTop(\mathcal{G}) \stackrel{\leftarrow}{\to} LTop(\mathcal{G}_0) : \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}

with Spec 𝒢 0 𝒢\mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}} left adjoint to the canonical functor p *p^* given by precomposition with pp.

Remark ( StSp p. 38 )

There is a canonical morphism

can:Pro(𝒢) opLTop(𝒢 0) can : Pro(\mathcal{G})^{op} \to LTop(\mathcal{G}_0)
Definition ( affine 𝒢\mathcal{G}-scheme, StSp 2.3.9)

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

Spec 𝒢:Pro(𝒢) opcanLTop(𝒢 0)Spec 𝒢 0 𝒢LTop(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\mathcal{G}-structured (∞,1)-topos in the image of this functor is an affine 𝒢\mathcal{G}-scheme.

𝒢\mathcal{G}-Schemes

Definition (geometric scheme, StSp 2.3.9)

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

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

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

such that

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

  • for every U iU_i there exists an equivalence

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

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

Definition (pregeometric scheme, StSp, 3.4.6)

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

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

Smooth 𝒢\mathcal{G}-schemes

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

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

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

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

With an envelope Τ𝒢\Tau \hookrightarrow \mathcal{G} fixed, a 𝒢\mathcal{G}-scheme is called smooth if there the affine schemes Spec 𝒢A i\mathbf{Spec}^{\mathcal{G}} A_i appearing in its definition may be chosen with A iA_i in the image of the includion τ𝒢\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 kk be a commutative ring. Recall the pregoemtry 𝒯 Zar(k)\mathcal{T}_{Zar}(k).

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

Derived smooth manifolds

Derived Deligne-Mumford stacks

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

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

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

Derived schemes with E 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

An indication of some details is in

See at E-∞ scheme and E-∞ geometry.

References

Generalized schemes are definition 2.3.9 of

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

Revised on February 15, 2014 14:52:37 by Urs Schreiber (89.204.139.210)