nLab
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

The notion of structured (,1)(\infty,1)-topos is a generalization of the notion of a locally ringed space and locally ringed topos generalized to (∞,1)-toposes. This is a way to formalize higher geometry/derived geometry-structure on little (∞,1)-toposs.

So a structured (,1)(\infty,1)-topos is an (∞,1)-topos 𝒳\mathcal{X} equipped with an (∞,1)-structure sheaf 𝒪\mathcal{O} that we think of as the collection of functions on 𝒳\mathcal{X} that preserve extra geometric structure – for instance continuous structure or smooth structure.

Being an \infty-function \infty-algebra, 𝒪(𝒰)\mathcal{O}(\mathcal{U}) is an algebra over an (∞,1)-algebraic theory 𝒢\mathcal{G}, called the (pre)geometry (for structured (∞,1)-toposes), since this encodes the nature of the extra geometric structure on 𝒳\mathcal{X}.

Formally therefore a geometric structure (,1)(\infty,1)-sheaf of 𝒳\mathcal{X} is a product/limit-preserving (∞,1)-functor

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

Here we think of 𝒳=Sh (,1)(C)\mathcal{X} = Sh_{(\infty,1)}(C) as being the (∞,1)-sheaf (∞,1)-topos on some (∞,1)-site CC and for any V𝒢V \in \mathcal{G} we think of

𝒪 VSh (,1)(C) \mathcal{O}_V \in Sh_{(\infty,1)}(C)

as being the (∞,1)-sheaf of structure-preserving functions on CC with values in VV.

Definition

Let SS be a geometry and let Sh(S)Sh(S) be the (∞,1)-topos of (∞,1)-sheaves on SS.

Notice that if S=Op(X)S = Op(X) is the nerve of the category of open subsets of some topological space XX, then Sh(X):=Sh(S)Sh(X) := Sh(S) is the (∞,1)-category of (∞,1)-sheaves on XX, as in the above motivating introduction.

We want to define a structure sheaf on SS (for instance on Op(X)Op(X)) of quantities modeled on some (,1)(\infty,1)-category VV to be an (∞,1)-functor

O X:VSh(S) O_X : V \to Sh(S)

to be thought of as the assignment to each value space vVv \in V of an (,1)(\infty,1)-sheaf of vv-valued functions on SS (on XX if S=Op(X)S = Op(X)).

But since we are taking care of the sheaf condition on SS, we also want to allow a similar kind of co-sheaf condition on VV. In order to do so, VV is taken to be equipped with extra structure encoding covers in VV, and O XO_X is then required to respect this structure suitably.

Geometries and admissibility structure

Definition

An admissiblility structure on an (,1)(\infty,1)-category VV is

  • a choice of sub (∞,1)-category V adVV^{ad} \hookrightarrow V, whose morphisms are to be called the admissible morphisms, such that

    • for every admissible morphism UXU \to X and any morphism XXX' \to X there is a diagram

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

      with UXU' \to X' admissible;

    • for every diagram of the form

      Y X Z \array{ && Y \\ & \nearrow && \searrow \\ X &&\to&& Z }

      with XZX \to Z and YZY \to Z admissible, also XYX \to Y is admissible.

  • a Grothendieck topology on VV which has the property that it is generated from a coverage consisting of admissible morphisms.

This is StrSh, def 1.2.1 in view of remark 1.2.4 below that.

Definition (StrSh, def 1.2.5)

An (,1)(\infty,1)-category VV equipped with an admissiblility structure is a geometry if it is essentially small, admits finite limits and is idempotent complete.

The admissible morphisms in an admissibility structure are roughly to be thought of as those morphisms that behave as open immersions ,

Structure sheaves: local \infty-algebras

Definition (StrSh, def 1.2.8)

(structure sheaf)

Let 𝒢\mathcal{G} be a geometry and 𝒳\mathcal{X} an (,1)(\infty,1)-topos An (∞,1)-functor

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

is a 𝒢\mathcal{G}-structure on 𝒳\mathcal{X} or 𝒢\mathcal{G}-structure sheaf on 𝒳\mathcal{X} if

  • it is a left exact (∞,1)-functor;

  • it respects gluing in 𝒢\mathcal{G} in that for {U iV} i\{U_i \to V\}_i a covering sieve consisting of admissible morphism, the induced morphism

    iO X(U i)O X(V) \coprod_i O_X(U_i) \to O_X(V)

    is an effective epimorphism in 𝒳\mathcal{X}.

Write Str 𝒢(𝒳)Func(𝒢,𝒳)Str_{\mathcal{G}}(\mathcal{X}) \subset Func(\mathcal{G},\mathcal{X}) for the full subcategory of such morphisms of the (∞,1)-category of (∞,1)-functors.

Remark

Without the condition on preservations of covers, the above defined the ∞-algebras over the essentially algebraic (∞,1)-theory 𝒢\mathcal{G}. The preservation of covers encodes the local 𝒢\mathcal{G}-algebras.

Therefore we shall equivalently write

𝒢Alg loc(𝒳)Str 𝒢(𝒳). \mathcal{G}Alg_{loc}(\mathcal{X}) \simeq Str_{\mathcal{G}}(\mathcal{X}) \,.
Remark

𝒢Alg loc(𝒳)\mathcal{G}Alg_{loc}(\mathcal{X}) is the (,1)(\infty,1)-category of algebras over an (,1)(\infty,1)-geometric theory.

This is discussed below.

As algebras over geometric (,1)(\infty,1)-theories

By the (∞,1)-Yoneda lemma, a cover-preserving functor 𝒪:𝒢𝒳\mathcal{O} : \mathcal{G} \to \mathcal{X} Yoneda extends equivalently to a (∞,1)-colimit-preserving (∞,1)-functor

𝒪:Sh (,1)(𝒢)𝒳. \mathcal{O} : Sh_{(\infty,1)}(\mathcal{G}) \to \mathcal{X} \,.

By the adjoint (∞,1)-functor theorem this has a right adjoint (∞,1)-functor and if 𝒪\mathcal{O} preserves finite (∞,1)-limits then so does its extension. Therefore local 𝒢\mathcal{G}-∞-algebras in 𝒳\mathcal{X} are equivalent to (∞,1)-geometric morphisms

𝒳𝒪Sh (,1)(𝒢). \mathcal{X} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to} Sh_{(\infty,1)}(\mathcal{G}) \,.

This means that structure sheaves 𝒪:𝒢𝒳\mathcal{O} : \mathcal{G} \to \mathcal{X} are equivalently encoded in geometric morphisms to the (∞,1)-category of (∞,1)-sheaves on the geometry.

Formally we have:

Proposition

For 𝒢\mathcal{G} a geometry, precomposition of the inverse image functor with the (∞,1)-Yoneda embedding y:𝒢Sh (,1)(𝒢)y : \mathcal{G} \to Sh_{(\infty,1)}(\mathcal{G}) induces an equivalence of (∞,1)-categories

Topos(𝒳,Sh (,1)(𝒢))𝒢Alg loc(𝒳) Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G})) \stackrel{\simeq}{\to} \mathcal{G}Alg_{loc}(\mathcal{X})

between the (∞,1)-category of (∞,1)-geometric morphisms from 𝒳\mathcal{X} to Sh (,1)(𝒢)Sh_{(\infty,1)}(\mathcal{G}) and the (,1)(\infty,1)-category of local 𝒢\mathcal{G}-∞-algebras in 𝒳\mathcal{X}.

This is (StrSp, prop 1.42).

Proof

This follows from the general fact, discussed in the section Local yoneda embedding at (∞,1)-Yoneda lemma that the essential image of the (,1)(\infty,1)-functor

Topos(𝒳,Sh (,1)(𝒢))LTopos(𝒳,PSh (,1)(𝒢))yFunc(𝒢,𝒳) Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G})) \stackrel{L}{\to} Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G})) \stackrel{y}{\to} Func(\mathcal{G}, \mathcal{X})

is spanned by the left exact and cover preserving functors.

Remark

We may think of this as saying that Sh (,1)(𝒢)Sh_{(\infty,1)}(\mathcal{G}) is the (,1)(\infty,1)-classifying topos for the (,1)(\infty,1)-geometric theory of local ∞-algebras over the essentially algebraic (∞,1)-theory 𝒢\mathcal{G}.

Local morphisms between structure sheaves

Remark

The (,1)(\infty,1)-category Str 𝒢(𝒳)Str_{\mathcal{G}}(\mathcal{X}) of 𝒢\mathcal{G}-structure sheaves on an (,1)(\infty,1)-topos 𝒳\mathcal{X} does not depend on the admissibility structure of 𝒢\mathcal{G}, but only on the Grothendieck topology induced by it.

(See StrSp, remark below prop. 1.4.2).

The admissibility structure does serve to allow the following definition of local morphisms of structure sheaves.

In terms of admissibility structures

Definition

(local morphism of structure sheaves)

A natural transformation η:𝒪𝒪:𝒢𝒳\eta : \mathcal{O} \to \mathcal{O}' : \mathcal{G} \to \mathcal{X} of structure sheaves is local if for every admissible morphism UXU \to X in 𝒢\mathcal{G} the naturality diagram

𝒪(U) η(U) 𝒪(U) 𝒪(X) η(X) 𝒪(X) \array{ \mathcal{O}(U) &\stackrel{\eta(U)}{\to}& \mathcal{O}'(U) \\ \downarrow && \downarrow \\ \mathcal{O}(X) &\stackrel{\eta(X)}{\to}& \mathcal{O}'(X) }

is a pullback square in 𝒳\mathcal{X}.

Write

Str 𝒢 loc(𝒳)Str 𝒢(𝒳) Str^{loc}_{\mathcal{G}}(\mathcal{X}) \subset Str_{\mathcal{G}}(\mathcal{X})

for the sub-(∞,1)-category of 𝒢\mathcal{G}-structures on 𝒳\mathcal{X} spanned by local transformations between them.

In terms of classifying (,1)(\infty,1)-toposes

Alternatively, the local transformations can be characterized as follows

it turns out the local transformations are the right half of a factorization system on Str 𝒢(𝒳)Str_{\mathcal{G}}(\mathcal{X}), and that this factorization system depends functorially on 𝒳\mathcal{X}, in that for every geometric morphism 𝒳𝒴\mathcal{X} \to \mathcal{Y} the induced Str 𝒢(𝒳)Str 𝒢(𝒴)Str_{\mathcal{G}}(\mathcal{X}) \to Str_{\mathcal{G}}(\mathcal{Y}) respects these factorization systems. (theorem 1.3.1)

This one can turn around, to characterize local transformations (and hence admissibility structures on 𝒢\mathcal{G}) in terms of functorial factorization systems on classifying (,1)(\infty,1)-toposes (def. 1.4.3):

For 𝒦\mathcal{K} an (,1)(\infty,1)-topos, declare that a geometric structure on 𝒦\mathcal{K} is a choice of factorization systems on Topos geom(𝒳,𝒦) opTopos_{geom}(\mathcal{X}, \mathcal{K})^{op} that is functorial in 𝒳\mathcal{X} . Given such we have another way of saying “local transformation”: this is the non-full subcategory Str 𝒦 loc(𝒳)Str^{loc}_{\mathcal{K}}(\mathcal{X}) of Topos geom(𝒳,𝒦) opTopos_{geom}(\mathcal{X}, \mathcal{K})^{op} on all objects and on the right part of the factorization system.

And this is indeed the same kind of datum as an admissibility structure on a geometry (example 1.4.4): in the case that 𝒦=Sh(𝒢)\mathcal{K} = Sh(\mathcal{G}) is the classifying topos for the geometry 𝒢\mathcal{G}, the defining equivalence Topos geom(𝒳,Sh(𝒢)) opStr 𝒢(𝒳)Topos_{geom}(\mathcal{X}, Sh(\mathcal{G}))^{op} \stackrel{\simeq}{\to} Str_{\mathcal{G}}(\mathcal{X}) identifies the two sub-categories of local transformations, Str 𝒢 loc(𝒳)Str^{loc}_{\mathcal{G}}(\mathcal{X}) and Str Sh(𝒢) loc(𝒳)Str^{loc}_{Sh(\mathcal{G})}(\mathcal{X}).

The (,1)(\infty,1)-category of structured (,1)(\infty,1)-toposes

Let (,1)Toposes(\infty,1)Toposes \subset (∞,1)Cat be the sub (∞,1)-category of (∞,1)-toposes: objects are (∞,1)-toposes, morphisms are geometric morphisms.

Write LTop:=(,1)Toposes opLTop := (\infty,1)Toposes^{op}.

Definition

((,1)(\infty,1)-category of 𝒢\mathcal{G}-structured (,1)(\infty,1)-toposes)

For 𝒢\mathcal{G} a geometry, the (,1)(\infty,1)-category of 𝒢\mathcal{G}-structured (,1)(\infty,1)-toposes

LTop(𝒢) LTop(\mathcal{G})

is defined as follows.

It is the sub (∞,1)-category

LTop(𝒢)Func(𝒢,ELTop)× Func(𝒢,LTop)LTop, LTop(\mathcal{G}) \subset Func(\mathcal{G}, E LTop) \times_{Func(\mathcal{G}, LTop)} LTop \,,

where ELTopLTopE LTop \to LTop is the coCartesian fibration associated by the (∞,1)-Grothendieck construction to the inclusion functor LTop(,1)CatLTop \hookrightarrow (\infty,1)Cat, spanned by the following objects and morphisms:

  • objects are 𝒢\mathcal{G}-structures 𝒪:𝒢𝒳\mathcal{O} : \mathcal{G} \to \mathcal{X} on some (,1)(\infty,1)-topos 𝒳\mathcal{X}:

    an object in Func(𝒢,ELTop)× Func(𝒢,LTop)LTopFunc(\mathcal{G}, E LTop) \times_{Func(\mathcal{G}, LTop)} LTop is an object 𝒳onLTop\mathcal{X} \on LTop together with a functor 𝒢ELTop 𝒳\mathcal{G} \to E LTop|_{\mathcal{X}} into the fiber of ETopE Top over that object; but that fiber is 𝒳\mathcal{X} itself, so an object in the fiber product is a functor 𝒢𝒳\mathcal{G} \to \mathcal{X} and this is in LTop(𝒢)LTop(\mathcal{G}) if it is a 𝒢\mathcal{G}-structure on 𝒳\mathcal{X};

  • morphisms α:𝒪𝒪\alpha : \mathcal{O} \to \mathcal{O}' are local morphisms of 𝒢\mathcal{G}-structures:

    for f *:𝒳𝒴f^* : \mathcal{X} \to \mathcal{Y} the image of α\alpha in LTopLTop, α\alpha is in LTop(𝒢)LTop(\mathcal{G}) precisely if for every admissible morphism UXU \to X in 𝒢\mathcal{G} the square

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

    is a pullback square in 𝒴\mathcal{Y}.

This is StrSp, def 1.4.8

The spectrum construction

For f:𝒢𝒢f : \mathcal{G} \to \mathcal{G}' a morphism of geometries, let

𝒪 𝒢 𝒢:Topos(𝒢)Topos(𝒢) \mathcal{O}_{\mathcal{G}}^{\mathcal{G}'} : Topos(\mathcal{G}') \to Topos(\mathcal{G})

be the induced functor on categories of structured toposes.

Theorem

This functor is a left adjoint (∞,1)-functor

(𝒪 𝒢 𝒢Spec 𝒢 𝒢)):Topos(𝒢)Spec 𝒢 𝒢𝒪 𝒢 𝒢Topos(𝒢). ( \mathcal{O}_{\mathcal{G}}^{\mathcal{G}'} \dashv Spec_{\mathcal{G}^{\mathcal{G}'}}) ) : Topos(\mathcal{G}) \stackrel{\overset{\mathcal{O}_{\mathcal{G}}^{\mathcal{G}'} }{\leftarrow}}{\underset{Spec_{\mathcal{G}}^{\mathcal{G}'} }{\to}} Topos(\mathcal{G}') \,.

This is (Lurie, theorem 2.1.1).

Definition

For 𝒢\mathcal{G} a geometry, let 𝒢 0\mathcal{G}_0 be the corresponding discrete geometry. We have a canonical morphism 𝒢 0𝒢\mathcal{G}_0 \to \mathcal{G}.

Write

Spec 𝒢:Pro(𝒢)Topos(𝒢 0)Spec 𝒢 0 mathTopos(𝒢) Spec^{\mathcal{G}} : Pro(\mathcal{G}) \to Topos(\mathcal{G}_0) \stackrel{Spec_{\mathcal{G}_0}^{\math}}{\to} Topos(\mathcal{G})

for the composite.

Theorem

This fits into an adjunction

(𝒪Spec):Pro𝒢Top(𝒢). (\mathcal{O} \dashv Spec) : Pro \mathcal{G} \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} Top(\mathcal{G}) \,.

This is (Lurie, theorem xyz).

Examples

Classes of examples

Canonical structure sheaves on objects in the big topos

For 𝒢\mathcal{G} a geometry let

H:=Sh (𝒢) \mathbf{H} := Sh_\infty(\mathcal{G})

be the (∞,1)-category of (∞,1)-sheaves on 𝒢\mathcal{G}. This is the big topos of higher geometry modeled on 𝒢\mathcal{G}. By the above discussion it is also the classifying topos of 𝒢\mathcal{G}-structure sheaves on toposes:

a 𝒢\mathcal{G}-valued structure sheaf 𝒪 𝒳:𝒢𝒳\mathcal{O}_{\mathcal{X}} : \mathcal{G} \to \mathcal{X} on an (∞,1)-topos 𝒳\mathcal{X} is equivalently an (∞,1)-geometric morphism

(p *p *):𝒳Hj𝒢 (p^* \dashv p_*) : \mathcal{X} \stackrel{\leftarrow}{\to} \mathbf{H} \stackrel{j}{\leftarrow} \mathcal{G}

in that 𝒪 𝒳=p *j\mathcal{O}_{\mathcal{X}} = p^* j, where jj is the (∞,1)-Yoneda embedding.

Notice that for every object XX \in \mathcal{H} its little topos-incarnation is the over-(∞,1)-topos H/X\mathbf{H}/X. This canonically sits over H\mathbf{H} by its etale geometric morphism

𝒳:=H/XX *X *H. \mathcal{X} := \mathbf{H}/X \stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}} \mathbf{H} \,.

So we have

Observation

The little topos 𝒳:=H/X\mathcal{X} := \mathbf{H}/X of every object XX in the big topos H\mathbf{H} over 𝒢\mathcal{G} is canonically equipped with a 𝒢\mathcal{G}-structure sheaf

𝒪 X:𝒢jHX *H/X. \mathcal{O}_X : \mathcal{G} \stackrel{j}{\to} \mathbf{H} \stackrel{X^*}{\to} \mathbf{H}/X \,.

We want to show that for V𝒢V \in \mathcal{G} the (∞,1)-sheaf 𝒪 X(V)\mathcal{O}_X(V) may indeed be thought of as the “sheaf of VV-valued functions on XX”.

Notice that for any VHV \in \mathbf{H} we have that X *(F)=(p 2:V×XX)X^*(F) = (p_2 : V \times X \to X).

Now assume first that XX is itself representable. Then by the discussion at over-(∞,1)-topos we have that H/X\mathbf{H}/X is a lucalization of PSh (𝒢)/XPSh (𝒢/X)PSh_\infty(\mathcal{G})/X \simeq PSh_\infty(\mathcal{G}/X), where 𝒢/X\mathcal{G}/X is the big site of XX. Under this equivalence (more details on this at over-topos) we have that (V×XX)(V \times X \to X) identifies with the presheaf given by

(UX)𝒢(U,V). (U \to X) \mapsto \mathcal{G}(U,V) \,.

This is the “sheaf of VV-valued functions on XX”.

(…)

Specific examples

Structure sheaves of continuous functions

Consider XX an ordinary topological space and Sh(X)Sh(X) the ordinary category of sheaves on its category of open subsets. Let 𝒢=Top\mathcal{G} = Top be some small version of Top with its usual Grothendieck topology with admissible covering families being open covers. Consider the functor

O X:TopSh(X) O_X : Top \to Sh(X)

that sends a topological space VV to the sheaf of continuous functions with values in VV:

O X(V):UC(U,V)=Hom Top(U,V). O_X(V) : U \mapsto C(U,V) = Hom_{Top}(U,V) \,.

By general properties of the hom-functor, this respects limits. The gluing condition says that for V 1,V 2VV_1, V_2 \subset V an open cover of VV by two patches, the morphism of sheaves

O X(V 1)O X(V 2)O X(V) O_X(V_1) \coprod O_X(V_2) \to O_X(V)

is an epimorphism of sheaves. This means that for each point xXx \in X the map of stalks

O X(V 1) xO X(V 2) xO X(V) x O_X(V_1)_x \coprod O_X(V_2)_x \to O_X(V)_x

is an epimorphism of sets. But this just says that given any function f:U xVf : U_x \to V on a neighbourhood U xU_x of xx, there is a smaller neighbourhood W xU xW_x \subset U_x such that the restriction f W xf|_{W_x} factors either through V 1V_1 or through V 1V_1. This is clearly the case by the fact that V 1,V 2V_1,V_2 form an open cover. (A neighbourhood of f(x)Vf(x) \in V exists which is contained in V 1V_1 or in V 2V_2, so take its preimage under ff as U xU_x).

Locally ringed spaces

StrSh, remark 2.5.11

Let XX be a topological space as before, but consider now the geometry 𝒢=CRing op\mathcal{G} = CRing^{op} to be the opposite category of commutative rings, where a covering family of SpecRCRing opSpec R \in CRing^{op} is a family of maps of the form RR[1r i]R \to R[\frac{1}{r_i}] with {r iR} i\{r_i \in R\}_i generating the unit ideal in RR. So we think of SpecR[1r i]Spec R[\frac{1}{r_i}] as an the open subset of SpecRSpec R on wich the function r ir_i does not vanish, and a covering family we think of as an open cover by such open subsets, in direct analogy to the above example.

Now given a sheaf of rings

O¯ XSh(X,CRing) \bar O_X \in Sh(X,CRing)

on XX (making XX a ringed space), which we may regard as the functor

O X:CRing opSh(X) O_X : CRing^{op} \to Sh(X)

that it represents

O X(SpecR):UHom CRing op(O¯ X(U),SpecR)=Hom CRing(R,O¯ X(U)) O_X(Spec R) : U \mapsto Hom_{CRing^{op}}(\bar O_X(U), Spec R) = Hom_{CRing}(R, \bar O_X(U))

we check what conditions this has to satisfy to qualify as a structure sheaf in the above sense.

The condition that

iO X(SpecR[1r i])SpecR \coprod_i O_X(Spec R[\frac{1}{r_i}]) \to Spec R

is an epimorphism of sheaves again means that it is stalkwise an epimorphism of sets. Now, a ring homomorphism R[1r i]O¯ X(U)R[\frac{1}{r_i}] \to \bar O_X(U) is given by a ring homomorphism f:RO X(U)f : R \to O_X(U) such that f(r i)f(r_i) is invertible in O X(U)O_X(U). (We think of this as the pullback of functions on SpecRSpec R to functions on UU by a map USpecRU \to Spec R that lands only in the open subset where the functoin r ir_i is non-vanishing).

So the condition that the above is an epimorphism on small enough UU says that for every ring homomorphism ϕ:RO¯ X(U)\phi : R \to \bar O_X(U) the value of ϕ\phi on at least one of the r ir_i is invertible element in O X(U)O_X(U).

Thinking of this dually this is just the same kind of statement as in the first example, really. But now we can say this more algebraically:

by assumption there is a linear combination of the r ir_i to the identity in RR

1= iα ir i 1 = \sum_i \alpha_i r_i

in RR (the partition of unity of functions on SpecRSpec R) and hence iα iϕ(r 1)=1\sum_i \alpha_i \phi(r_1) = 1 in (O X) x(O_X)_x That for this invertible finite sum at least one of the summands is invertible is the condition that (O X) x(O_X)_x is a local ring .

So a ringed space has a structure sheaf in the above sense if it is a locally ringed space.

Ordinary ringed spaces

It may be worthwhile to retell the motivating example in the “Idea” introduction above for the maybe more familiar case of ordinary (1-categorical) structure sheaves with values in (unital) rings.

An ordinary topological space XX with its category of open subsets Op(X)Op(X) is a ringed space or Op(X)Op(X) is a ringed site if it is equipped with a sheaf O X:Op(X) opRingsO_X : Op(X)^{op} \to Rings with values in the category of rings. For UXU \subset X one thinks of O X(U)O_X(U) as the ring of allowed functions on UU.

If for the moment we ignore the technicality that O XO_X is supposed to be a sheaf and just regard it as a presheaf, and if we furthermore invoke the idea of space and quantity and think of a ring RR as a generalized quantity in form of a copresheaf, canonically the co-representable co-presheaf

R:(Ring fin) opSet R : (Ring^fin)^{op} \to \Set

on finitely generated rings, which sends

R:RHom(R,R) R : R' \mapsto Hom(R,R')

then we find that O XO_X is in fact a presheaf on Op(X)Op(X) with values in a co-presheaf on (Ring fin) op(Ring^{fin})^{op}

O X:Op(X) op[(Ring fin) op,Set] O_X : Op(X)^{op} \to [(Ring^{fin})^{op}, Set]

or equivalently a generalized quantity on (Ring fin) op(Ring^{fin})^{op} with values in presheaves on XX:

O X:(Ring fin) op[Op(X) op,Set]. O_X : (Ring^{fin})^{op} \to [Op(X)^{op}, Set] \,.

Since rings can be identified with left-exact functors (Ring fin) opSet(Ring^{fin})^{op}\to Set, we don’t need to impose any admissibility structure in order to recover the notion of a sheaf of rings, since left-exactness is part of the definition of a “structure sheaf.” We do, however, need an admissibility structure if we want to recover the notion of a sheaf of local rings, as in the previous example above.

Derived ringed spaces

Now formulate the previous example according to the above definition:

Let CRing finCRing^{fin} be the category of finitely generated commutative rings There is a standard admissibility structure on (CRing fin) op(CRing^{fin})^{op} that makes it a geometry in the above sense.

Then for XX a topological space an (,1)(\infty,1)-functor (CRing fin) opSh(S)(CRing^{fin})^{op} \to Sh(S) to (infty,1)(infty,1)-sheaves on XX is a sheaf of local commutative rings on XX. (StrSh, example 1.2.13)

To generalize this to derived structure sheaves we replace the category of rings here with the (,1)(\infty,1)-category of simplicial rings.

Definition (StrSh def 4.1.1)

The (,1)(\infty,1)-category of simplicial commutative rings over an ordinary commutative ring kk is

SCR k:=PSh Σ(FreeAlg k) SCR_k := PSh_\Sigma(FreeAlg_k)

the (,1)(\infty,1)-category of (∞,1)-presheaves on commutative kk-algebras of the form k[x 1,,x n]k[x_1, \cdots, x_n].

Then

Derived smooth manifolds

(StrSh, example 4.5.2)

Every ordinary smooth manifold XX becomes canonically a generalized space with structure sheaf as follows:

Let V:=DiffV := Diff be some version of the category of smooth manifolds. This becomes a pregeometry in the above sense by taking admissible morphisms to be inclusions of open submanifolds.

Then for Sh(X):=Sh(Op(X))Sh(X) := Sh(Op(X)) the (,1)(\infty,1)-topos of (,1)(\infty,1)-sheaves on XX, the obvious (,1)(\infty,1)-functor

O X:VSh(X) O_X : V \to Sh(X)

which for every co-test manifold vv is the sheaf

O X(v):(UX)Hom Diff(U,v) O_X(v) : (U \subset X) \mapsto Hom_{Diff}(U,v)

is a DiffDiff-structure sheaf. Notice that this is precisely nothing but the structure sheaf of smooth functions from the introduction above.

The point is that there are other, more fancy structure sheaves

O X:VSh(X) O_X : V \to Sh(X)

possible. They describe derived smooth manifolds as described in DerSmooth.

Properties

Limits and colimits

Let 𝒢\mathcal{G} be a geometry for structured (infinity,1)-toposes. Write F:(,1)Topos(𝒢)F : (\infty,1)Topos(\mathcal{G}) \to (∞,1)Topos for the forgetful (∞,1)-functor from 𝒢\mathcal{G}-structured (,1)(\infty,1)-toposes to their underlying (,1)(\infty,1)-topos.

Proposition

The (,1)(\infty,1)-category (,1)Topos(𝒢)(\infty,1)Topos(\mathcal{G}) has all cofiltered (∞,1)-limits and the forgetful functor F:(,1)Topos(𝒢)(,1)ToposF : (\infty,1)Topos(\mathcal{G}) \to (\infty,1)Topos preserves these.

This appears as (Lurie, corl 1.5.4).

Embedding into the ambient big (,1)(\infty,1)-topos

Definition

For 𝒢\mathcal{G} a geometry (for structured (∞,1)-toposes) write

𝒢^:=Pro(𝒢) \hat \mathcal{G} := Pro(\mathcal{G})

for its (∞,1)-category of pro-objects.

Write Grpd^\widehat{\infty Grpd} for the very large (∞,1)-category of large ∞-groupoids and

Sh^(𝒢^,Grpd^) \hat Sh(\hat \mathcal{G}, \widehat{\infty Grpd})

for the very large (∞,1)-sheaf (∞,1)-topos.

Proposition

The canonical inclusion

Scheme(𝒢)PSh^(𝒢^,Grpd^) Scheme(\mathcal{G}) \to \hat PSh(\hat \mathcal{G}, \widehat{\infty Grpd})

of locally representable structured (∞,1)-toposes by

(X,𝒪 X)Hom(Spec(),(X,𝒪 X)) (X, \mathcal{O}_X) \mapsto Hom(Spec(-), (X, \mathcal{O}_X))

is a full and faithful (∞,1)-functor.

This is (Lurie, theorem, 2.4.1).

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

References

The notion of structured (,1)(\infty,1)-toposes was introduced in

Analogous precursor discussion in 1-category theory, hence for ringed toposes is in

  • Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).

The special case of “smoothly structured spaces” (derived smooth manifolds) is discussed in

Revised on June 12, 2013 02:58:26 by Urs Schreiber (82.169.65.155)