Spahn sheaf on a sheaf

Motivation

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

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

where

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

• $L$ sends a sheaf on $X$ to its espace étale.

Très petit topos

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

We assume that $fet$ is the admissible class defined by an infinitesimal modality $\Box$ on $H$.

Definition (universal $G$-structure, classifying topos)

(1) A $G$-structure $O$ on an $(\infty,1)$-topos is called universal if for every $(\infty,1)$-topos $X$ composition with $O$ induces an equivalence of $(\infty,1)$-categories

$$Fun^*(K,T)\to Str_G(T)$$

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

(2) In this case we say $O$ exhibits $K$ as classifying $(\infty,1)$-topos for $G$-structures on $X$.

Remark

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

Proof

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

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

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

Local representability of the très petit topos

Relative-, and absolute spectra

Definition (pro objects)

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

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

and the hom sets are

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

We have (more or less) synonyms:

• pro object, cofiltered, having a cone

• ind object, filtered, having a cocone

Definition 2.2.2 (admissible morphism in $Pro(H)$)

Let $H$ be a geometry. A morphism $U\to X$ in $Pro(H)$ is called $Pro(H)$- admissible if it arises as a pullback

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

in $Pro(H)$ where $f:U^\prime \to X^\prime$ is admissible in $H$.

Remark 2.2.7

$Pro(H)^{ad}/X$ is essentially small, hence the presheaf- and sheaf category on it can be defined in the usual way.

Definition (2.2.9 $Spec_X$ as a $H$-structured topos)

Let $H$ be a geometry, let $X\in H$ be an object. Then we define the $H$-stuctured topos $(Spec X, O_{Spec X})$ by:

$$Spec X:=Sh(Pro(H)^{ad}/ X)$$

$$O_{Spec X}:=(H\stackrel{\tilde{O}_{Spec X}}{\to} Psh(Pro(H)^{ad} /X)\stackrel{L}{\to} Spec X:=Sh(Pro(H)^{ad}/X))$$

where $L$ is sheafification (left adjoint to the inclusion and

$$\tilde{O}_{Spec X}:=(H\times (Pro(H)^{ad}/X)^{op}\to H\times Pro(H)^{op}=Lex(H,\infty Grpd)\stackrel{ev}{\to} \infty Grpd$$

Proof

We have to show that $O_{Spec X}$ is a $H$-structure on $Spec X$.

Lemma 2.2.14

Let $H$ be a geometry, let $X\in Proj(H)$ a pro object of $H$. Let $j:Pro(H)^{ad}/X \to Spec X$ denote the composite of the top arrows, let $proj$ denote the projection in

$$\array{ &&Pro(H)^{ad}/X&\stackrel{Y}{\hookrightarrow} Psh(Pro(H)^{ad}/X)\stackrel{Sh(-)}{\to} Sh(Pro(H)^{ad}/X)=Spec X \\ &&\downarrow^{proj}&&\nearrow^{Lan_{proj} j} \\ H&\stackrel{y}{\hookrightarrow}& Pro(H) }$$

and let $Y$ and $y$ denote Yoneda embeddings. Then we have the equivalence

$$((Lan_{proj} j)\circ y)\simeq O_{Spec X}$$

Restriction of $H$-structured $(\infty,1)$-toposes, $H$-schemes

Digression (DAG V, Prop.2.3.7)

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

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

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

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

$$f_{|U_i}:((X/U_i,(O_{G,X})_{|U_i})\to (Y,O_{G,Y})$$

is étale, then $f$ is étale.

Definition (Restriction 2.3.3, DAG chapter 2.3)

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

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

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

$$G\stackrel{O_{G,X}}{\to}X\stackrel{p^*}{\to}X/U$$

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

Definition ($LTop$, $LTop(G)$)

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

(2) For a geometry $G$ we identify with $LTop(G)$ the $(\infty,1)$-category which has as objects $G$-structures $O:G\to H$ on some $(\infty,1)$-topos $H$ and morphisms are those natural transformations $f^*$ 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 $p:G\to G_0$ be a morphism of geometries. Let $p^*:=(-)\circ p:LTop(G_0)\to L Top(G)$ the restriction functor.

(1) Then there is an adjunction

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

where the left adjoint is called a relative spectrum functor.

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

$$Spec_G:= Spec_{G,G_0}\circ \iota$$

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

A construction of the absolute spectrum (DAGV section 2.2)

Remark (The $H$-structured global section functor $\Gamma_H$)

Let $p:\overline{LTop}\to LTop$ be the universal topos fibration. We interpret objects of $\overline{LTop}$ as pairs $(X,x)$ where $X$ is a topos and $x\in X$ is an object of $X$.

(…)

The global section functor $\Gamma:\overline{LTop}\to\infty Grpd$ factors through $Lex(H,\infty Grpd)\simeq Pro(H)^{op}$. (…) The resulting map $\Gamma_H:LTop(H)\to Pro(H)^{op}$ is called $H$-structured global section functor.

Proposition (Theorem 2.2.12, $(\Spec_H\dashv \Gamma_H)$)

Let $H$ be a geometry, let $X$ be an object of $Pro(G)$.

(1) Then

$$(Spec_H\dashv \Gamma_H):LTop(H)\stackrel{\Gamma_H}{\to}Pro_H$$

is an adjunction.

(2) $O_Y$ factors as

$$H\stackrel{j}{\to}Pro (H)\stackrel{\overline{O_Y}}{\to} Y$$

where $j$ denotes the Yoneda embedding and $\overline{O_Y}$ preserves small limits. And we have the identification of mapping spaces

$$Pro(H)^{op}(X,\Gamma_H(Y,O_Y))\simeq Y(*, \overline{O_Y}(X))$$

Digression (Example 2.3.8)

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

$$Spec_H U\to Spec_H X$$

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 $f:U\to X$ be admissible in $Pro (H)$. Then $Spec_H$ preserves finite limits, hence we can assume that $f$ arises from an admissible morphism $f_0:U_U\to X_0$ in $H$.

Let $Spec_H X=:(\chi, O_\chi)$ such that $O_\chi(X_0)$ has a canonical global section $\eta:1_\chi\to O_\chi(X_0)$. Let $Y:=hfiber(\eta, O_\chi(f_0))$ and let

$$(\Upsilon,O_\Upsilon):=(\chi/Y, O_\chi |Y)$$

Then there is a canonical global section $\eta^\prime$ of $O_\Upsilon(U_0)$.

This means that $\eta^\prime$ exhibits $(\Upsilon,O_\Upsilon)$ as an absolute spectrum $U$ such that $Spec_H f$ can be identified with the étale map $(\Upsilon, O_\Upsilon)\to (\chi,O_\chi)$.

Definition

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

• there exists a collection $\{U_i \in X\}$

such that

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

• for every $U_i$ there exists an equivalence

  $$(X/{U_i}, O_{G,X}|_{U_i}) \simeq Spec_{G} A_i$$

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

Remark

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

Lemma

If (cond) admissible morphisms in the essential image of the Yoneda embedding $(H/X)^{fet}\hookrightarrow Pro((H/X)^{fet})$ are preserved (pulled back to formally étale morphisms) by pullback we have:

$$\array{ (H/X)^{fet}/U&\simeq &Sh((H/X)^{fet})/U\\ &\simeq&Sh(Pro(H/X)^{ad}/U)&:=& Spec U}$$

where the first equivalence holds since $(H/X)^{fet}$ is a topos, the second one is implied by DAGV Notation 2.2.2 and Remark 2.2.7 and (cond), the $:=$ is definition 2.2.9. wrt. the geometry $(H/X)$

Remark

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

Proof

$O:H\to E$ has to satisfy

• $O$ is left exact

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

The terminal object in $E:=(H/X)^{fet}$ is $id_X$ the identity on $X$. The collection of all formally étale effective epimorphisms (in $H$) with codomain $X$ covers $X$. By HTT Remark 6.2.3.6. they cover $id_X$ in the slice.

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

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

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

The restriction $(E/U,O_{| U})$ of $(E,O)$ to $U$ is given by:

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

• The restriction of the $H$-structure $O$ is given as follows:

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

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

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

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

$$(Spec_{H_0,H}\dashv p^*):LTop(H_0)\stackrel{Spec_{H_0,H}}{\to} LTop(H)$$

and the absolute spectrum $Spec_H$ is defined to be the composit

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

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