Spahn sheaf on a sheaf (Rev #14, changes)

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Motivation

Let XHX\in H be a space (an object of a category HH of spaces), let Sh(X)Sh(X) be the category of sheaves on the frame of opens on XX, let (H/X) et(H/X)^{et} denote the wide subcategory of H/XH/X with only étale morphisms. Then there is an adjoint equivalence

(LΓ):(H/X) etΓSh(X)(L\dashv \Gamma):(H/X)^{et}\stackrel{\Gamma}{\to}Sh(X)

where

  • Γ\Gamma sends an étale morphism f:UXf:U\to X to the sheaf of local sections of ff.

  • LL sends a sheaf on XX to its espace étale.

Très petit topos

We wish to clarify in which sense also the (,1)(\infty,1)- topos (H/X) fet(H/X)^{fet} can be regarded as an (,1)(\infty,1)-sheaftopos on XX. One formulation of this is to show that ((H/X) fet,O (H/X) fet)((H/X)^{fet},O_{(H/X)^{fet}}) is a locally representable structured (,1)(\infty,1)-topos - and that the representation is exhibited by formally étale morphisms.

We assume that fetfet is the admissible class defined by an infinitesimal modality \Box on HH.

Definition (universal GG-structure, classifying topos)

(1) A GG-structure OO on an (,1)(\infty,1)-topos is called universal if for every (,1)(\infty,1)-topos XX composition with OO induces an equivalence of (,1)(\infty,1)-categories if

Fun *(K,T)Str G(T)Fun^*(K,T)\to Str_G(T)

where Fun *(K,T)Fun^*(K,T) denotes the geometric morphisms ff with inverse image f *:KTf^*:K\to T.

(2) In this case we say OO exhibits KK as classifying (,1)(\infty,1)-topos for GG-structures on XX.

Remark

HH, H/XH/X, and (H/X) fet(H/X)^fet are (H/X) fet(H/X)^{fet}-structured (,1)(\infty,1)-toposes.

Proof

The classifying topos for (G/X) fet(G/X)^{fet}-structures is HH and the (,1)(\infty,1)-toposes in question are linked with HH by geometric morphisms. We obtain the required structures as the image of

Fun *(H,H/X)Str (H/X) fet(H/X)Fun^*(H,H/X)\to Str_{(H/X)^{fet}}(H/X)

respectively for HH and (H/X) fet(H/X)^{fet} in place of “H/X”.

Local representability of the très petit topos

Definition (pro objects)

Let CC be an (,1)(\infty,1)-category. We have Ind(C op)Pro(C) opInd(C^{op})\simeq Pro(C)^{op}. A pro object in in CC is a formal limit of a cofiltered diagram in CC. A cofiltered diagram is defined to be a finite diagram FF having a cone (i.e. a family of natural transformation κ cF\kappa_c\to F for all cCc\in C, where κ c\kappa_c denotes the constant functor having value cc). So we have

Pro(C)={F:DC|Disfinite,cofiltered}Pro(C)=\{F:D\to C | D\,is\,finite,\,cofiltered\}

and the hom sets are

Pro(C)(F,G)=lim eEcolim dDC(F(d),G(e))Pro(C)(F,G)=lim_{e\in E}colim_{d\in D}C(F(d),G(e))

We have (more or less) synonyms:

  • pro object, cofiltered, having a cone

  • ind object, filtered, having a cocone

Digression (DAG V, Prop.2.3.7)

(1) A morphisms f:(X,O G,X)(Y,O G,Y)f:(X,O_{G,X})\to (Y,O_{G,Y}) is called étale if (1a) the underlying geometric morphism of (,1)(\infty,1)-toposes is étale and (1b) the induced map f *:XYf^*:X\to Y is an equivalence in Str G(𝔘)Str_G(\mathfrak{U})

(2) Condition (1b) is equivalent to the requirement that ff is pp-cocartesian for p:LTop(G)LTopp:LTop(G)\to LTop the projection.

(3) Being an étale geometric morphism of structured (,1)(\infty,1)-toposes is a local property:

If there is an effective epimorphism iU i* X\coprod_i U_i\to *_X to the terminal object of XX, and f:(X,O G,X)(Y,O G,Y)f:(X,O_{G,X})\to (Y,O_{G,Y}) in LTop(G) opLTop(G)^{op} a morphism such that

f |U i:((X/U i,(O G,X) |U i)(Y,O G,Y)f_{|U_i}:((X/U_i,(O_{G,X})_{|U_i})\to (Y,O_{G,Y})

is étale, then ff is étale.

Definition (Restriction 2.3.3, DAG chapter 2.3)

Let (X,O G,X)(X,O_{G,X}) be a structured (,1)topos,let (\infty,1)-topos, (\infty,1) let U\in -topos, X$ let be an object.UXU\in X be an object.

(1) The restriction of XX to UU is defined to be the slice X/UX/U.

(2) The restriction (O G,X) |U(O_{G,X})_{| U} of O G,XO_{G,X} to UU is defined to be composite

GO G,XXp *X/UG\stackrel{O_{G,X}}{\to}X\stackrel{p^*}{\to}X/U

where p *p^* is base change along p:U*p:U\to *.

Definition (relative- and absolute spectrum)

Let p:GG 0p:G\to G_0 be a morphism of geometries. Let p *:=()p:LTop(G 0)LTop(G)p^*:=(-)\circ p:LTop(G_0)\to L Top(G) the restriction functor.

(1) Then there is an adjunction

(Spec G,G 0p *):LTop(G 0)p *LTop(G)(Spec_{G,G_0}\dashv p^*):L Top(G_0)\stackrel{p^*}{\to}LTop(G)

where the left adjoint is called a relative spectrum functor.

(2) Let now G 0G_0 be the discrete geometry underlying GG. Then

Spec G:=Spec G,G 0ιSpec_G:= Spec_{G,G_0}\circ \iota

is called absolute spectrum functor; here ι:Ind(G op)LTop(G 0)\iota:Ind(G^{op})\hookrightarrow LTop(G_0) denotes the inclusion of the ind objects of GG.

Definition

A GG-structured (∞,1)-topos (X,O G,X)(X,O_{G,X}) is called locally representable (aka a GG-scheme) if

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

such that

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

  • for every U iU_i there exists an equivalence

    (X/U i,O G,X| U i)Spec GA i (X/{U_i}, O_{G,X}|_{U_i}) \simeq Spec_{G} A_i

    of structured (,1)(\infty,1)-toposes for some A iPro(G)A_i \in Pro(G) (in the (∞,1)-category of pro-objects in GG). In other words (X,O G,X)(X,O_{G,X}) is assumed to be locally equivalent to an absolute spectrum (aka affine scheme) of a pro object in GG.

Remark

((H/X) fet,O)((H/X)^{fet},O)If is locally representable (C,O C)(C,O_C)HH is a structured topos and -structured (C/U,(O C) |U)(C/U,(O_C)_{|U})(,1)(\infty,1) is an restriction thereof, then -topos.(C,O C)(C/U,(O C) |U)(C,O_C)\to (C/U,(O_C)_{|U}) is an étale morphism of structured toposes.

Proof Remark

The terminal object in ((H/X) fet,O)((H/X)^{fet},O)E:=(H/X) fetE:=(H/X)^{fet} is locally representable is HHid Xid_X-structured the identity on (,1)(\infty,1)XX-topos.. The collection of all formally étale effective epimorphisms (in HH) with codomain XX covers XX and hence the cover id Xid_X in the slice.

We know that EHE\hookrightarrow H is a reflective- and coreflective subcategory of HH (if not argue OO is leftexact, then an adjoint functor theorem implies that EE is coreflective subcat.). We think of O=rO=r being the reflector

Let Uid XU\to id_X be an element of the cover; i.e. a formally étale effective epimorphism UXU\to X. Since EE is a coreflective subcategory O(U)O(X)O(U)\to O(X) is a cover of O(X)O(X).

The restriction (E/U,(O E) |U)(E/U,(O_E)_{| U}) of (E,O E)(E,O_E) is given by:

  • Objects of E/UE/U are cocones A X U \array{A&\to &X\\\searrow &&\swarrow\\&U&} where AXA\to X is formally étale. Morphisms are pyramids with four faces and tip UU.

  • The restriction of the HH-structure O:=O EO:=O_E is given as follows:

HOEp *H/UH\stackrel{O}{\to}E\stackrel{p^*}{\to}H/U

where p *p^* is base change along p:U*p:U\to *. Pro objects in EE are cofiltered diagrams in EE or -equivalently filtered diagrams in E opE^{op}

Proof

The terminal object in E:=(H/X) fetE:=(H/X)^{fet} is id Xid_X the identity on XX. The collection of all formally étale effective epimorphisms (in HH) with codomain XX covers XX and hence the cover id Xid_X in the slice. This follows from HTT Remark 6.2.3.6.

We know that EH/XE\hookrightarrow H/X is a reflective- and coreflective subcategory of HH. H/XHH/X\to H is a geometric morphism. Hence we can choose O:HEO:H\to E to be a left adjoint which preserves effective epimorphisms.

Let Uid XU\to id_X be an element of the cover; i.e. a formally étale effective epimorphism UXU\to X. Hence O(U)O(X)O(U)\to O(X) is a cover of O(X)O(X).

The restriction (E/U,(O E) |U)(E/U,(O_E)_{| U}) of (E,O E)(E,O_E) is given by:

  • Objects of E/UE/U are cocones A X U \array{A&\to &X\\\searrow &&\swarrow\\&U&} where AXA\to X is formally étale. Morphisms are pyramids with four faces and tip UU.

  • The restriction of the HH-structure O:=O EO:=O_E is given as follows:

HOEp *H/UH\stackrel{O}{\to}E\stackrel{p^*}{\to}H/U

where p *p^* is base change along p:U*p:U\to *. Pro objects in EE are cofiltered diagrams in EE or -equivalently filtered diagrams in E opE^{op}

Revision on December 17, 2012 at 04:39:16 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.