Spahn sheaf on a sheaf (Rev #19, changes)

Showing changes from revision #18 to #19: Added | ~~Removed~~ | ~~Chan~~ged

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 if

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 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 (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 $G$ .

Definition Digression (Example 2.3.8)

A Let $G H$ G H ~~-structured~~ ~~(∞,1)-topos~~ be a geometry, let $(f : U \to X, O_{G, X})$ ~~(X,O_{G,X})~~ f:U\to X is be ~~called~~ an admissible morphism in~~locally representable~~ $Pro(H)$ . ~~(aka~~ Then a ~~$G$ -scheme) if~~

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

Spec_H U\to Spec_H X

~~such~~ is ~~that~~ an étale morphism of absolute spectra.

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 Proof

If This follows from the following Proposition (Theorem 2.2.12). ~~$(C,O_C)$~~ ~~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.~~

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 $(\Upsilon, O_\Upsilon)\to (\chi,O_\chi)$ .

Remark Definition

~~There~~ A ~~exists~~ a $H G$ H G ~~-structure~~ -structured (∞,1)-topos $O (X, O_{G, X})$ O (X,O_{G,X}) on is called ~~$(H/X)^{fet}$~~ locally representable ~~such~~ (aka ~~that~~ a ~~$((H/X)^{fet},O)$~~ $G$ -scheme ) is if a ~~locally~~ ~~representable~~ ~~$H$~~ ~~-structured~~ ~~$(\infty,1)$~~ ~~-topos.~~

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 $G$ .

Proof Remark

~~$O:H\to E$~~ If ~~has to satisfy~~ $(C,O_C)$ 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.

$O$ 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 $(E/U,(O_E)_{| U})$ 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:

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

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 17:09:33 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn sheaf on a sheaf (Rev #19, changes)

Motivation

Très petit topos

Definition (universal GG-structure, classifying topos)

Remark

Proof

Local representability of the très petit topos

Definition (pro objects)

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)

Definition Digression (Example 2.3.8)

Remark Proof

Remark Definition

Proof Remark

Remark

Proof

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

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