Spahn sheaf on a sheaf (Rev #15, changes)

Showing changes from revision #14 to #15: 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 (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

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

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

Proof

~~The terminal object in~~ $O:H\to E$ ~~$E:=(H/X)^{fet}$~~ has to satisfy is ~~$id_X$~~ ~~the identity on~~ ~~$X$~~ ~~. The collection of all formally étale effective epimorphisms (in~~ ~~$H$~~ ~~) with codomain~~ ~~$X$~~ ~~covers~~ ~~$X$~~ ~~and hence the cover~~ ~~$id_X$~~ ~~in the slice. This follows from HTT Remark 6.2.3.6.~~

We know that $E\hookrightarrow H/X$ is a reflective- and coreflective subcategory of $H$ . $H/X\to H$ is a geometric morphism. Hence we can choose $O:H\to E$ to be a left adjoint which preserves effective epimorphisms.

$O$ is left exact
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 $E$ .

~~Let~~ The terminal object in $U E \to : {id}_{X} = (H / X)^{fet}$ ~~U\to~~ E:=(H/X)^{fet} ~~id_X~~ be is an ~~element~~ of ~~the~~ ~~cover;~~ ~~i.e.~~ a ~~formally~~ ~~étale~~ ~~effective~~ ~~epimorphism~~ $U {id}_{X} \to X$ ~~U\to~~ id_X X . ~~Hence~~ the identity on $O (U) \to O (X)$ ~~O(U)\to~~ X ~~O(X)~~ . is The a collection ~~cover~~ of all formally étale effective epimorphisms (in $O H (X)$ ~~O(X)~~ H . ) with codomain $X$ covers $X$ . By HTT Remark 6.2.3.6. they cover $id_X$ in the slice. This follows from

~~The~~ Now ~~restriction~~ we choose $(O E : / = U r, \circ (p^{*} O_{E})_{| U})$ ~~(E/U,(O_E)_{|~~ O:=r\circ ~~U})~~ p^* to be the composit of base change $(b^{*} E :, H O_{E} \to) H / X$ ~~(E,O_E)~~ b^*:H\to H/X is along ~~given~~ ~~by:~~ $b:X\to *$ (this functor is exact) 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.

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 $O:=O_E$ is given as follows:

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

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

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

~~where~~ The restriction $p^{*} (E / U, (O_{E})_{| U})$ p^* (E/U,(O_E)_{| U}) is of ~~base~~ ~~change~~ ~~along~~ $p (: E U, \to O *)$ ~~p:U\to~~ (E,O) * . ~~Pro~~ to ~~objects~~ in $E U$ E U ~~are~~ is ~~cofiltered~~ given ~~diagrams~~ by: in ~~$E$~~ ~~or -equivalently filtered diagrams in~~ ~~$E^{op}$~~

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 $p:U\to *$ . Pro objects in $E$ are cofiltered diagrams in $E$ or -equivalently filtered diagrams in $E^{op}$

Revision on December 17, 2012 at 05:34:18 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn sheaf on a sheaf (Rev #15, 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 (relative- and absolute spectrum)

Definition

Remark

Remark

Proof

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