Spahn sheaf on a sheaf (Rev #20)

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

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 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 (Remark 2.2.7)

Let HH be a geometry. Then every admissible morphism UXU\to X in Pro(H)Pro(H) arises as a pullback

U j(U ) X j(X )\array{ U&\to&j(U^\prime) \\ \downarrow&&\downarrow \\ X&\to&j(X^\prime) }

in Pro(H)Pro(H).

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)(\infty,1)-topos, let 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 (LTopLTop, LTop(G)LTop(G))

(1) LTopLTop is defined to be the (,1)(\infty,1)-category which objects are (,1)(\infty,1)-toposes and which morphisms are geometric morphisms of (,1)(\infty,1)-toposes ff such that the inverse image f *f^* preserves small colimits and finite limits.

(2) For a geometry GG we identify with LTop(G)LTop(G) the (,1)(\infty,1)-category which has as objects GG-structures O:GHO:G\to H on some (,1)(\infty,1)-topos HH and morphisms are those natural transformations f *f^* which (1.) are inverse images of geometric morphisms and (2.) whose naturality square is a pullback square in every admissible morphism.

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.

Proposition (Theorem 2.2.12, (Spec HΓ H)(\Spec_H\dashv \Gamma_H))

Let HH be a geometry, let XX be an object of Pro(G)Pro(G).

(1) Then

(Spec HΓ H):LTop(H)Γ HPro H(Spec_H\dashv \Gamma_H):LTop(H)\stackrel{\Gamma_H}{\to}Pro_H

is an adjunction.

(2) O YO_Y factors as

HjPro(H)O Y¯YH\stackrel{j}{\to}Pro (H)\stackrel{\overline{O_Y}}{\to} Y

where jj denotes the Yoneda embedding and O Y¯\overline{O_Y} preserves small limits. And we have the identification of mapping spaces

Pro(H) op(X,Γ H(Y,O Y))Y(*,O Y¯(X))Pro(H)^{op}(X,\Gamma_H(Y,O_Y))\simeq Y(*, \overline{O_Y}(X))
Digression (Example 2.3.8)

Let HH be a geometry, let f:UXf:U\to X be an admissible morphism in Pro(H)Pro(H). Then

Spec HUSpec HXSpec_H U\to Spec_H X

is an étale morphism of absolute spectra.

Proof

This follows from the previous Proposition (Theorem 2.2.12).

A direct proof goes as follows: Let f:UXf:U\to X be admissible in Pro(H)Pro (H). Then Spec HSpec_H preserves finite limits, hence we can assume that ff arises from an admissible morphism f 0:U UX 0f_0:U_U\to X_0 in HH.

Let Spec HX=:(χ,O χ)Spec_H X=:(\chi, O_\chi) such that O χ(X 0)O_\chi(X_0) has a canonical global section η:1 χO χ(X 0)\eta:1_\chi\to O_\chi(X_0). Let Y:=hfiber(η,O χ(f 0))Y:=hfiber(\eta, O_\chi(f_0)) and let

(ϒ,O ϒ):=(χ/Y,O χ|Y)(\Upsilon,O_\Upsilon):=(\chi/Y, O_\chi |Y)

Then there is a canonical global section η \eta^\prime of O ϒ(U 0)O_\Upsilon(U_0).

This means that η \eta^\prime exhibits (ϒ,O ϒ)(\Upsilon,O_\Upsilon) as an absolute spectrum UU such that Spec HfSpec_H f can be identified with the étale map (ϒ,O ϒ)(χ,O χ)(\Upsilon, O_\Upsilon)\to (\chi,O_\chi).

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

If (C,O C)(C,O_C) is a structured topos and (C/U,(O C) |U)(C/U,(O_C)_{|U}) is an restriction thereof, then (C,O C)(C/U,(O C) |U)(C,O_C)\to (C/U,(O_C)_{|U}) is an étale morphism of structured toposes.

Remark

There exists a HH-structure OO on (H/X) fet(H/X)^{fet} such that ((H/X) fet,O)((H/X)^{fet},O) is a locally representable HH-structured (,1)(\infty,1)-topos.

Proof

O:HEO:H\to E has to satisfy

  • OO is left exact

  • OO satisfies codescent: For every collection of admissible (i.e. formally étale) morphisms {U iX}\{U_i\to X\} in HH which generate a covering sieve on XX, the induced map iO(U i)O(X)\coprod_i O(U_i)\to O(X) is an effective epimorphism in EE.

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. By HTT Remark 6.2.3.6. they cover id Xid_X in the slice.

Now we choose O:=rp *O:=r\circ p^* to be the composit of base change (this functor is exact) b *:HH/Xb^*:H\to H/X along b:X*b:X\to * followed by the coreflector (that we have a coreflector is shown (reference)) r:H/XEr:H/X\to E (this functor is right adjoint and hence left exact). In total OO is left exact and since our cover consists only of formally étale morphisms rr and hence OO preserve the cover.

Now we describe the restriction of (E,O)(E,O) to an element UU of the cover:

Let Uid XU\to id_X be an element of the cover; i.e. a formally étale effective epimorphism UXU\to X.

The restriction (E/U,O |U)(E/U,O_{| U}) of (E,O)(E,O) to UU 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 OO is given as follows:

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

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

Now we show that (E,O)(E, O) is locally equivalent to an absolute spectrum:

Let H 0H_0 denote the discrete geometry (admissible morphisms are precisely all equivalences) with underlying category HH. Let h:H 0Hh:H_0\to H be a morphism of geometries (i.e. hh preserves finite limits, maps admissible morphism to such, the image of an admissible cover is an admissible cover). Then there is an adjunction

(Spec H 0,Hp *):LTop(H 0)Spec H 0,HLTop(H)(Spec_{H_0,H}\dashv p^*):LTop(H_0)\stackrel{Spec_{H_0,H}}{\to} LTop(H)

and the absolute spectrum Spec HSpec_H is defined to be the composit

Ind(H op)Pro(H) opLex(H,Grpd)LTop(H 0)Spec H 0,HLTop(H)Ind(H^{op})\simeq Pro(H)^{op}\simeq Lex(H, \infty Grpd)\hookrightarrow LTop(H_0)\stackrel{Spec_{H_0,H}}{\to}LTop(H)

(Pro objects in HH are cofiltered diagrams in HH or -equivalently - filtered diagrams in H opH^{op})

Revision on December 17, 2012 at 18:03:24 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.