Spahn sheaf on a sheaf (Rev #21, changes)

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Motivation

Let X∈HX\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

(L⊣Γ):(H/X) et→ΓSh(X)(L\dashv \Gamma):(H/X)^{et}\stackrel{\Gamma}{\to}Sh(X)

where

  • Γ\Gamma sends an étale morphism f:U→Xf: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 *:K→Tf^*: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 c→F\kappa_c\to F for all c∈Cc\in C, where κ c\kappa_c denotes the constant functor having value cc). So we have

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

and the hom sets are

Pro(C)(F,G)=lim e∈Ecolim d∈DC(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

Relative-, and absolute spectra
Digression Definition (Remark (pro 2.2.7) objects)

Let H C H C be a an geometry. Then every admissible morphismU(→∞X,1) U\to (\infty,1) X -category. in We have Pro Ind(HC op)≃Pro(C) op Pro(H) Ind(C^{op})\simeq Pro(C)^{op} . arises A as pro a object pullback inCC 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 c→F\kappa_c\to F for all c∈Cc\in C, where κ c\kappa_c denotes the constant functor having value cc). So we have

U → j(U ′) ↓ ↓ X → j(X ′)Pro(C)={F:D→C|Disfinite,cofiltered} \array{ Pro(C)=\{F:D\to U&\to&j(U^\prime) C \\ | \downarrow&&\downarrow D\,is\,finite,\,cofiltered\} \\ X&\to&j(X^\prime) }

in and the hom sets arePro(H)Pro(H).

Pro(C)(F,G)=lim e∈Ecolim d∈DC(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 (Definition V, 2.2.2) Prop.2.3.7)

(1) Let A morphisms f H:(X,O G,X)→(Y,O G,Y) f:(X,O_{G,X})\to H (Y,O_{G,Y}) is be called a étale geometry. if A (1a) the underlying geometric morphism of(U∞→,X1) (\infty,1) U\to X -toposes is inétalePro(H)Pro(H) and is (1b) called the induced mapf *Pro : ( X H → )Y f^*:X\to Pro(H) Y - is admissible an if equivalence it in arises as a pullbackStr 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.

U → j(U ′) ↓ ↓ j(f) X → j(X ′)\array{ U&\to&j(U^\prime) \\ \downarrow&&\downarrow^{j(f)} \\ X&\to&j(X^\prime) }

(3) in Being an étale geometric morphism of structuredPro(∞H,1) (\infty,1) Pro(H) -toposes is where a local property:f:U ′→X ′f:U^\prime \to X^\prime is admissible in HH.

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 Digression (Restriction (DAG 2.3.3, V, DAG Prop.2.3.7) chapter 2.3)

Let (1) A morphismsf:(X,O G,X)→(Y,O G,Y) (X,O_{G,X}) f:(X,O_{G,X})\to (Y,O_{G,Y}) be is a called structured étale if (1a) the underlying geometric morphism of(∞,1)(\infty,1) -topos, -toposes let isU∈XU\in Xétale be and an (1b) object. the induced mapf *:X→Yf^*:X\to Y is an equivalence in Str G(𝔘)Str_G(\mathfrak{U})

(1) (2) The Condition restriction (1b) of is equivalent to the requirement that X f X f to is U p U p -cocartesian is for defined to be the slice X p / : U LTop(G)→LTop X/U p:LTop(G)\to LTop . the projection.

(2) (3) The Being restriction an étale geometric morphism of structured(O G,X∞) |U,1) (O_{G,X})_{| (\infty,1) U} -toposes of is a local property:O G,XO_{G,X} to UU is defined to be composite

G→O G,XX→p *X/UG\stackrel{O_{G,X}}{\to}X\stackrel{p^*}{\to}X/U

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

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

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)LTopLTop, LTop(G)LTop(G))

(1) LetLTop(X,O G,X) LTop (X,O_{G,X}) is defined to be the a structured(∞,1)(\infty,1) -category -topos, which let objects are(U∞∈,X1) (\infty,1) U\in X -toposes and be which an morphisms object. are geometric morphisms of(∞,1)(\infty,1)-toposes ff such that the inverse image f *f^* preserves small colimits and finite limits.

(2) (1) For The a restriction geometry of G X G X we to identify with LTop U(G) LTop(G) U is defined to be the slice(X∞/,U1) (\infty,1) X/U -category . which has as objectsGG-structures O:G→HO: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.

(2) The restriction (O G,X) |U(O_{G,X})_{| U} of O G,XO_{G,X} to UU is defined to be composite
G→O G,XX→p *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)LTopLTop, LTop(G)LTop(G))

Let (1) p LTop:G→G 0 p:G\to LTop G_0 is defined to be a the morphism of geometries. Letp *:=(−∞ ) ,∘1p:LTop(G 0)→LTop(G) p^*:=(-)\circ (\infty,1) p:LTop(G_0)\to L Top(G) -category the which restriction objects functor. 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.

(1) (2) Then For there a is geometry an adjunctionGG we identify with LTop(G)LTop(G) the (∞,1)(\infty,1)-category which has as objects GG-structures O:G→HO: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.

(Spec G,G 0⊣p *):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 0∘ιSpec_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 Definition (Theorem (relative- 2.2.12, and absolute spectrum)(Spec H⊣Γ H)(\Spec_H\dashv \Gamma_H))

Let H p:G→G 0 H p:G\to G_0 be a geometry, morphism let of geometries. LetXp *:=(−)∘p:LTop(G 0)→LTop(G) X p^*:=(-)\circ p:LTop(G_0)\to L Top(G) be the an restriction object functor. ofPro(G)Pro(G).

(1) Then there is an adjunction

(Spec HG,G 0⊣Γ Hp *):LTop(G 0)→p *LTop( H G)→Γ HPro H (Spec_H\dashv (Spec_{G,G_0}\dashv \Gamma_H):LTop(H)\stackrel{\Gamma_H}{\to}Pro_H p^*):L Top(G_0)\stackrel{p^*}{\to}LTop(G)

where the left adjoint is an called adjunction. arelative spectrum functor.

(2) Let now O G Y0 O_Y G_0 factors be as the discrete geometry underlyingGG. Then

HSpec G→j:Pro=(Spec G,G 0H∘)ι→O Y¯Y H\stackrel{j}{\to}Pro Spec_G:= (H)\stackrel{\overline{O_Y}}{\to} Spec_{G,G_0}\circ Y \iota

where is calledjjabsolute spectrum functor ; denotes here the Yoneda embedding andO Y¯ι:Ind(G op)↪LTop(G 0) \overline{O_Y} \iota:Ind(G^{op})\hookrightarrow LTop(G_0) preserves denotes small limits. And we have the identification inclusion of mapping the spaces ind objects ofGG.

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:U→Xf:U\to X be an admissible morphism in Pro(H)Pro(H). Then

Spec HU→Spec HXSpec_H U\to Spec_H X

is an étale morphism of absolute spectra.

A construction of the absolute spectrum (DAGV section 2.2)
Proof Remark (TheHH-structured global section functor Γ H\Gamma_H)

This Let follows from the previous Proposition (Theorem 2.2.12).p:LTop¯→LTopp:\overline{LTop}\to LTop be the universal topos fibration. We interpret objects of LTop¯\overline{LTop} as pairs (X,x)(X,x) where XX is a topos and x∈Xx\in X is an object of XX.

A (…) direct proof goes as follows: Letf:U→Xf: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 U→X 0f_0:U_U\to X_0 in HH.

Let The global section functorSpec HΓX=:(LTop¯χ→,∞O χGrpd) Spec_H \Gamma:\overline{LTop}\to\infty X=:(\chi, Grpd O_\chi) such factors that throughO χLex(X 0H,∞Grpd)≃Pro(H) op O_\chi(X_0) Lex(H,\infty Grpd)\simeq Pro(H)^{op} . has (…) a The canonical resulting global map sectionηΓ H:1 χLTop→O χ(X 0H)→Pro(H) op \eta:1_\chi\to \Gamma_H:LTop(H)\to O_\chi(X_0) Pro(H)^{op} . Let is calledY:=hfiber(η,O χ(f 0))Y:=hfiber(\eta, O_\chi(f_0))HH-structured global section functor . 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 Proposition (Theorem 2.2.12,(Spec H⊣Γ H)(\Spec_H\dashv \Gamma_H))

A Let G H G H -structured (∞,1)-topos be a geometry, let(X,O G,X) (X,O_{G,X}) X is be called an object oflocally representablePro(G)Pro(G) . (aka aGG-scheme) if

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

(1) Then

such that

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

  • 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 i∈Pro(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.

is an adjunction.

(2) O YO_Y factors as
H→jPro(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))
Remark Digression (Example 2.3.8)

If Let(HC,O C) (C,O_C) H is be a structured geometry, topos let and(fC:/U , →(XO C) |U) (C/U,(O_C)_{|U}) f:U\to X is be an restriction admissible thereof, morphism then inPro( C H,O C)→(C/U,(O C) |U) (C,O_C)\to Pro(H) (C/U,(O_C)_{|U}) . is Then an étale morphism of structured toposes.

Spec HU→Spec HXSpec_H U\to Spec_H X

is an étale morphism of absolute spectra.
Remark Proof

There This exists follows a from the previous Proposition (Theorem 2.2.12).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.

A direct proof goes as follows: Let f:U→Xf: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 U→X 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 χ→O 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).
Proof Definition

O:H→EO:H\to EA has to satisfyGG-structured (∞,1)-topos (X,O G,X)(X,O_{G,X}) is called locally representable (aka a GG-scheme) if

  • there exists a collection

    OO is left exact

    {U i∈X}\{U_i \in X\}

  • OO satisfies codescent: For every collection of admissible (i.e. formally étale) morphisms {U i→X}\{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 such terminal that object inE:=(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:=r∘p *O:=r\circ p^* to be the composit of base change (this functor is exact) b *:H→H/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/X→Er: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.

  • 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 i∈Pro(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.

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

Let U→id XU\to id_X be an element of the cover; i.e. a formally étale effective epimorphism U→XU\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 A→XA\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:
H→OE→p *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 0→Hh: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,H⊣p *):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) op≃Lex(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})
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:H→EO:H\to E has to satisfy

  • OO is left exact

  • OO satisfies codescent: For every collection of admissible (i.e. formally étale) morphisms {U i→X}\{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:=r∘p *O:=r\circ p^* to be the composit of base change (this functor is exact) b *:H→H/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/X→Er: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 U→id XU\to id_X be an element of the cover; i.e. a formally étale effective epimorphism U→XU\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 A→XA\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:

H→OE→p *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 0→Hh: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,H⊣p *):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) op≃Lex(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 20:39:40 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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