etale geometric morphism


Topos Theory

topos theory



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Topos morphisms

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In higher category theory




A sheaf FF on (the site of open subsets of) a topological space XX corresponds to an étalé space π F:Y FX\pi_F : Y_F \to X. This space Y FY_F has itself a sheaf topos associated to it, and the map Y FXY_F \to X induces a geometric morphism of sheaf toposes

π F:Sh(Y F)Sh(X). \pi_F : Sh(Y_F) \to Sh(X) \,.

Due to the special nature of Y FY_F, the topos on the left is equivalent to the slice topos Sh(X)/FSh(X)/F, and the projection morphism above factors through a canonical standard geometric morphism Sh(X)/FSh(X)Sh(X)/F \to Sh(X)

π F:Sh(Y F)Sh(X)/FSh(X). \pi_F : Sh(Y_F) \stackrel{\simeq}{\to} Sh(X)/F \to Sh(X) \,.

And conversely, every local homeomorphism YXY \to X of topological spaces corresponds to a geometric morphism of sheaf toposes of this form.

This motivates calling a geometric morphism

𝒳𝒴 \mathcal{X} \to \mathcal{Y}

a local homeomorphism of toposes or étale geometric morphism if it factors as an equivalence followed by a projection out of an overcategory topos.

If the topos is a locally ringed topos, or more generally a structured (∞,1)-topos, it makes sense to require additionally that the local homeomorphism is compatible with the extra structure.


For H\mathbf{H} a topos (or (∞,1)-topos, etc.) and for XHX \in \mathbf{H} an object, the overcategory H /X\mathbf{H}_{/X} is also a topos ((,1)(\infty,1)-topos, etc), the slice topos (slice (∞,1)-topos, …).

The canonical projection π !:H /XH\pi_! : \mathbf{H}_{/X} \to \mathbf{H} is part of an essential (in fact, locally connected/ locally ∞-connected) geometric morphism:

π=(π !π *π *):H /Xπ *π *π !H. \pi = (\pi_! \dashv \pi^* \dashv \pi_*) : \mathbf{H}_{/X} \stackrel{\overset{\pi_!}{\longrightarrow}}{\stackrel{\overset{\pi^*}{\longleftarrow}}{\underset{\pi_*}{\longrightarrow}}} \mathbf{H} \,.

This is the base change geometric morphism for the terminal morphism X*X \to *.

For toposes


A geometric morphism KH\mathbf{K} \to \mathbf{H} is called a local homeomorphism of toposes, or an étale geometric morphism, if it is equivalent to such a projection— in other words, if it factors by geometric morphisms as KH /XπH\mathbf{K} \stackrel{\simeq}{\to} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H} for some XHX \in \mathbf{H} .

For structured toposes

If the (∞,1)-toposes in question are structured (∞,1)-toposes, then this is refined to the following


A morphism f:(𝒳,𝒪 𝒳)(𝒴,𝒪 𝒴)f : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}}) of structured (∞,1)-toposes is an étale morphism if

  1. the underlying morphism of (,1)(\infty,1)-toposes is an étale geometric morphism;

  2. the induced map f *𝒪 𝒴𝒪 𝒳f^* \mathcal{O}_\mathcal{Y} \to \mathcal{O}_\mathcal{X} is an equivalence.

This is StSp, Def. 2.3.1.


If H\mathbf{H} is a localic topos Sh(S)Sh(S) over a topological space SS we have that XSh(S)X \in Sh(S) corresponds to an étalé space over XX and H/XH\mathbf{H}/X \to \mathbf{H} to an étale map.

If 𝒢\mathcal{G} is a geometry (for structured (∞,1)-toposes) then for f:UXf : U \to X an admissible morphism in 𝒢\mathcal{G}, the induced morphism of structured (∞,1)-toposes

Spec 𝒢USpec 𝒢X Spec^\mathcal{G} U \to Spec^{\mathcal{G}} X

is an étale geometric morphism of structured (,1)(\infty,1)-toposes.

This is StrSp, example 2.3.8.



The inverse image of an étale geometric morphism is a cartesian closed functor.

See at cartesian closed functor for proof.



An étale geometric morphism is a cartesian Wirthmüller context.


(recognition of étale geometric morphisms)

A geometric morphism (f *f *):KH(f^* \dashv f_*) : \mathbf{K} \to \mathbf{H} is étale precisely if

  1. it is essential;

  2. f !f_! is a conservative functor;

  3. For every diagram XYf !ZX \to Y \leftarrow f_! Z in H\mathbf{H} the induced diagram

    f !(f *X× f *YZ) f !Z X Y \array{ f_!(f^* X \times_{f^* Y} Z) &\to& f_! Z \\ \downarrow && \downarrow \\ X &\to& Y }

    is a pullback diagram.

For (∞,1)-toposes this is HTT, prop.


(Recovering a topos from its etale overcategory)

For H\mathbf{H} an (,1)(\infty,1)-topos we have

H((,1)Topos/H) et, \mathbf{H} \simeq ((\infty,1)Topos/\mathbf{H})_{et} \,,

where ((,1)Topos/H) et(,1)Topos/H((\infty,1)Topos/\mathbf{H})_{et} \subset (\infty,1)Topos/\mathbf{H} is the full sub-(∞,1)-category of the over-(∞,1)-category on the etale geometric morphisms KH\mathbf{K} \to \mathbf{H}.

This is HTT, remark


The notion of local homeomorphisms of toposes is page 651 (chapter C3.3) of

The notion of étale geometric morphisms between (∞,1)-toposes is introduced in section 6.3.5 of

Discussion of the refinement to structured (∞,1)-toposes is in section 2.3 of

Revised on January 20, 2015 21:46:24 by Urs Schreiber (