Spahn sheaf on a sheaf (Rev #20)

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 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

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

Digression (Remark 2.2.7)

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

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

in $Pro(H)$ .

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 $G$ .

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.

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 $H^{op}$ )

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.

Spahn sheaf on a sheaf (Rev #20)

Motivation

Très petit topos

Definition (universal GG-structure, classifying topos)

Remark

Proof

Local representability of the très petit topos

Definition (pro objects)

Digression (Remark 2.2.7)

Digression (DAG V, Prop.2.3.7)

Definition (Restriction 2.3.3, DAG chapter 2.3)

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

Definition (relative- and absolute spectrum)

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

Digression (Example 2.3.8)

Proof

Definition

Remark

Remark

Proof

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

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

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