nLab
geometry (for structured (infinity,1)-toposes)

Context

Higher geometry

(,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

Contents

Idea

A geometry 𝒢\mathcal{G} is an (∞,1)-category equipped in a compatible way with

  1. the structure of an (∞,1)-site;

  2. the structure of an essentially algebraic (∞,1)-theory.

The objects of 𝒢\mathcal{G} are to be thought of as test-spaces with certain higher geometry structure and the morphisms as homomorphisms preserving that geometric structure.

These two structures gives rise to

  1. The big (∞,1)-topos Sh(𝒢)Sh(\mathcal{G}) of (∞,1)-sheaves on 𝒢\mathcal{G}. Its objects are generalized spaces given by rules

    X:𝒢 opX : \mathcal{G}^{op} \to ∞Grpd

    for how to map test spaces into them.

  2. The (∞,1)-algebras over 𝒢\mathcal{G} in some little topos 𝒳\mathcal{X}, given by rules

    𝒪:𝒢𝒳 \mathcal{O} : \mathcal{G} \to \mathcal{X}

    that send each obect U𝒢U\in \mathcal{G} to a UU-valued structure sheaf.

Using the additional structure of a site on 𝒢\mathcal{G} allows to identify those structure sheaves 𝒪\mathcal{O} that are local in that they respect coverings. This constitutes a generalized notion of locally ringed toposes called 𝒢\mathcal{G}-structured (∞,1)-toposes. Equivalently these local structure sheaves are given by (∞,1)-geometric morphisms 𝒳𝒪H=Sh(𝒢) \mathcal{X} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to} \mathbf{H} = Sh(\mathcal{G}) to the big topos over 𝒢\mathcal{G}.

Definition

Geometry

Summary

A geometry on an (∞,1)-category 𝒢\mathcal{G} is a Grothendieck topology on 𝒢\mathcal{G} together with

  • the extra structure given the information of which covering morphisms are to be thought of as local homeomorphisms

  • the extra property that it has all finite limits.

If only all finite products exist we speak of a pre-geometry. Every pregeometry 𝒯\mathcal{T} extends uniquely 𝒯𝒢\mathcal{T} \hookrightarrow \mathcal{G} to an enveloping geometry 𝒢\mathcal{G}.

When the objects of the geometry 𝒢\mathcal{G} are thought of as test spaces (affine schemes), the objects of the pregeometry 𝒯𝒢\mathcal{T} \hookrightarrow \mathcal{G} are to be thought of as the affine spaces. This distinction is used to encode smoothness of maps between test spaces: a morphism in 𝒢\mathcal{G} is smooth if it locally factors through admissible maps between objects in 𝒯\mathcal{T}.

Definition (admissibility structure, StSp 1.2.1)

An admissibility structure on an (∞,1)-category 𝒢\mathcal{G} is a Grothendieck topology on 𝒢\mathcal{G} that is generated from its intersection with a subcategory 𝒢 ad𝒢\mathcal{G}^{ad} \subset \mathcal{G} whose morphisms – called the admissible morphisms have the following properties

  • admissible morphisms are stable under (∞,1)-pullback;

  • admissible morphisms satisfy “left cancellability”, meaning that whenever in

    Y f g X h Z \array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z }

    gg and hh are admissible, then so is ff.

  • admissible morphisms are closed under retracts.

Equivalently, this is a Grothendieck topology on 𝒢\mathcal{G} which is generated from admissible morphisms.

Admissibility

As will become clear when looking at examples, the notion of admissible morphisms models the idea of maps between test spaces that behave like open injections or, more generally, as local homeomorphisms .

Definition ( Structured Spaces 1.2.5 )

A geometry (for (,1)(\infty,1)-toposes) is

Definition (discrete geometry, StSp, 1.2.10)

The discrete geometry 𝒢 0\mathcal{G}^0 on 𝒢\mathcal{G} is given by

  • the admissible morphisms in 𝒢\mathcal{G} are precisely the equivalences

  • the Grothendieck topology on 𝒢\mathcal{G} is trivial: a sieve is covering only if it is maximal.

Every small (∞,1)-category CC becomes a geometry by regarding it as a discrete geometry in the above way.

Pregeometry

Definition (pregeometry, StSp, 3.1.1)

A pregeometry (for structured (∞,1)-toposes) is

  • an (∞,1)-category 𝒯\mathcal{T};

  • equipped with an admissibility structure (homotopical topology)

such that

  • 𝒯\mathcal{T} has all products.
Remark

So a geometry differs from a pregeometry in that it is idempotent complete and closed not only under products but under all finite limits.

Various concepts for geometries have immediate analogues for pregeometries.

Smooth morphisms

Definition (smooth morphism, Structured Spaces, 3.1.7)

A morphism f:XSf : X \to S in a pregeometry 𝒯\mathcal{T} is called smooth if it is locally stably admissible in that there exists a cover {u i:U iX}\{u_i : U_i \to X\} (meaning: generators of a covering sieve) of XX by admissible morphisms, such that on U iU_i the morphism ff factors admissibly through some S×V iS \times V_i in that there is a commuting diagram

U i u i X S×V i p 1 S. \array{ U_i &\stackrel{u_i}{\to}& X \\ \downarrow && \downarrow \\ S \times V_i &\stackrel{p_1}{\to}& S } \,.
Remark

To interpret this, recall that we think of admissible morphisms as injections of open subsets.

Proposition (Structured Spaces, 3.1.8)
  • Smooth morphisms are stable under pullback.

  • pregeometric 𝒯\mathcal{T}-structures 𝒪:𝒯𝒳\mathcal{O} : \mathcal{T} \to \mathcal{X} preserve pullbacks of smooth morphisms.

Structured (,1)(\infty,1)-topos

Definition (geometry structure on an (,1)(\infty,1)-topos, StSp, 1.2.8)

For 𝒢\mathcal{G} a geometry, and TSh (S)T \simeq Sh_\infty(S) an (∞,1)-topos, a 𝒢\mathcal{G}-structure on the (,1)(\infty,1)-topos TT making it a structured (∞,1)-topos is a (∞,1)-functor

C():𝒢T C(-) : \mathcal{G} \to T

such that

  • C()C(-) is left exact (preserves finite limits);

  • C()C(-) satisfies codescent (the dual notion of descent): for π:(V= iV i)W\pi : (V = \coprod_i V_i) \to W any cover by admissible morphisms in GG, the induced morphism

    C(π):C(V)C(W) C(\pi) : C(V) \to C(W)

    is an effective epimorphism in TT, i.e. its Čech nerve is a simplicial resolution of C(W)C(W):

    Čech(C(π))C(W). \mathop{Čech}(C(\pi)) \stackrel{\simeq}{\to} C(W) \,.
Definition (pregeometry structure on an (,1)(\infty,1)-topos, StrSp 3.1.4)

Let 𝒯\mathcal{T} be a pregeometry and 𝒳\mathcal{X} an (∞,1)-topos.

A 𝒯\mathcal{T}-structure on 𝒳\mathcal{X} is an (∞,1)-functor 𝒪:𝒯𝒳\mathcal{O} : \mathcal{T} \to \mathcal{X} such that

  • 𝒪\mathcal{O} preserves finite products.

  • 𝒪\mathcal{O} preserves pullbacks of admissible morphism in that for every pullback diagram

    U U f X X \array{ U' &\to& U \\ \downarrow && \downarrow^f \\ X' &\to& X }

    in 𝒯\mathcal{T} with ff admissible, the image

    𝒪(U) 𝒪(U) f 𝒪(X) 𝒪(X) \array{ \mathcal{O}(U') &\to& \mathcal{O}(U) \\ \downarrow && \downarrow^f \\ \mathcal{O}(X') &\to& \mathcal{O}(X) }

    is again a pullback.

  • 𝒪\mathcal{O} respects covers by admissible morphisms in that for every covering sieve {f i:U iX}\{f_i : U_i \to X\} in 𝒯\mathcal{T} by admissible f if_i the induced map i𝒪(U i)𝒪(X)\coprod_i \mathcal{O}(U_i) \to \mathcal{O}(X) is an effective epimorphism in 𝒳\mathcal{X}.

Remark

The first clause says that 𝒪:𝒯𝒳\mathcal{O} : \mathcal{T} \to \mathcal{X} is in particular an \infty-algebra over the (multi-sorted) (∞,1)-algebraic theory 𝒯\mathcal{T}.

The other two clauses encode that this \infty-algebra 𝒪\mathcal{O} indeed behaves like a function algebra .

Definition (geometric envelope, StrSp 3.4.1)

…the universal geometry extending a pregeometry

Proposition (StSp 3.4.5)

Let 𝒯\mathcal{T} be a pregeometry and f:𝒯𝒢f : \mathcal{T} \to \mathcal{G} a morphism that exhibits the geometry 𝒢\mathcal{G} as a geometric envelope of 𝒯\mathcal{T}. Then for every (∞,1)-topos 𝒳\mathcal{X} precomposition with ff induces an equivalence of (∞,1)-categories of 𝒯\mathcal{T}- and 𝒢\mathcal{G}-structures on 𝒳\mathcal{X}:

Str 𝒢(𝒳)Str 𝒯(𝒳),Str 𝒢(𝒳) locStr 𝒯(𝒳) loc Str_{\mathcal{G}}(\mathcal{X}) \stackrel{\simeq}{\to} Str_{\mathcal{T}}(\mathcal{X}) \,,\;\;\;\; Str_{\mathcal{G}}(\mathcal{X})^{loc} \stackrel{\simeq}{\to} Str_{\mathcal{T}}(\mathcal{X})^{loc}

Examples

Étale geometry

If we regard the ordinary étale site as a pregeometry 𝒯 et\mathcal{T}_{et}, then its geometric envelope 𝒢 et\mathcal{G}_{et} is the étale (∞,1)-site. See derived étale geometry for the precise statement

Proposition

The 1-localic 𝒢 et\mathcal{G}_{et}-generalized schemes are precisely Deligne-Mumford stacks (without the separation axiom).

See Deligne-Mumford stack for details.

Smooth geometry

There should be a geometry 𝒢\mathcal{G} such that 𝒢\mathcal{G}-generalized schemes are precisely derived smooth manifolds.

Analogous structures in the axiomatic context of differential cohesion are discussed in differential cohesion – structure sheaves.

References

The general theory is developed in

The definition of a geometry 𝒢\mathcal{G} is def. 1.2.5.

A 𝒢\mathcal{G}-structure on an (∞,1)-topos is in def. 1.2.8.

The notion of 𝒢\mathcal{G}-spectrum – which are (∞,1)-toposes – is the subject of section 2.1 .

The inclusion

Spec 𝒢:𝒢Str(𝒢) Spec^{\mathcal{G}} : \mathcal{G} \hookrightarrow Str(\mathcal{G})

is definition 2.1.2.

The definition of 𝒢\mathcal{G}-generalized scheme is definition 2.3.9, page 51.

The inclusion

Sch(𝒢)Sh (Ind(𝒢)) Sch(\mathcal{G}) \hookrightarrow Sh_\infty(Ind(\mathcal{G}))

is the topic of section 2.4, theorem 2.4.1

The special case of “smoothly structured spaces” called derived smooth manifold is discussed in

Apart from looking at the special case this article also contains useful introduction and details on the general case.

In the version of this that is available on the arXiv (arXiv) the point of view is more on bi-presheaves, a useful discussion to the relation to structured morphisms here is in section 10.1 there.

Revised on December 18, 2012 14:32:58 by Urs Schreiber (131.174.40.67)