nLab
skeletal geometric morphism

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Topos Theory

topos theory

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Extra stuff, structure, properties

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In higher category theory

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Idea

Skeletal geometric morphisms are those geometric morphisms that preserve double negation sheaves and therefore play a role in the descriptions of classes of toposes like e.g. Boolean or De Morgan toposes in whose definitions the negation participates.

The notion of a skeletal geometric morphism can be viewed as a weakening of the notion of open geometric morphism.

Definition

A geometric morphism f:f : \mathcal{F} \to \mathcal{E} is called skeletal if the following equivalent conditions hold:

  • ff restricts to a geometric morphism Sh ¬¬()Sh ¬¬()Sh_{\neg\neg}(\mathcal{F}) \to Sh_{\neg\neg}(\mathcal{E}) .

  • The inverse image f *f^\ast preserves ¬¬\neg\neg-dense monomorphisms.

  • For any subobject AAA'\rightarrowtail A in \mathcal{E}: ¬¬f *(A)=¬f *(¬A)\neg\neg f^\ast(A')=\neg f^\ast(\neg A') in Sub (f *(A))Sub_\mathcal{F}(f^\ast(A)).

Example

  • Inverse images of open geometric morphisms are Heyting functors, hence commute with ¬\neg and, therefore, open geometric morphisms are skeletal. In particular, geometric morphisms with Boolean codomain are open (Johnstone 2002, p.612), hence skeletal.

  • Dense subtoposesi:Sh j()i:Sh_j(\mathcal{E})\hookrightarrow \mathcal{E} are precisely those subtoposes with Sh ¬¬(Sh j())=Sh ¬¬()Sh_{\neg\neg}(Sh_j(\mathcal{E}))=Sh_{\neg\neg}(\mathcal{E}) (cf. this proposition) and, therefore, are skeletal.

Properties

The following two propositions concern skeletal inclusions (cf. Johnstone (2002, p.1007)):

Proposition

An inclusion i:Sh j()i:Sh_j(\mathcal{E})\hookrightarrow \mathcal{E} is skeletal iff ext(j)ext(j) , the jj-closure of 010\rightarrowtail 1 , is a ¬¬\neg\neg-closed subterminal object of \mathcal{E}.

Proof: First, notice that in general for a topology jj a subterminal UU is jj-closed iff it is a jj-sheaf since 11 is always a jj-sheaf. Hence 00 is ¬¬\neg\neg-closed precisely when it is a ¬¬\neg\neg-sheaf.

Now assume ii skeletal. Since ext(j)ext(j) is a ¬¬\neg\neg-sheaf in Sh j()Sh_j(\mathcal{E}) and ii preserves them, it is also a ¬¬\neg\neg-sheaf in \mathcal{E}.

Conversely, assume ext(j)ext(j) is a ¬¬\neg\neg-sheaf in \mathcal{E}. Since it is also a jj-sheaf, it is contained in Sh j()Sh ¬¬()Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E}) but this coincides with Sh ¬¬(Sh j())Sh_{\neg\neg}(Sh_j(\mathcal{E})) because as a subtopos of the Boolean Sh ¬¬()Sh_{\neg\neg}(\mathcal{E}) the intersection Sh j()Sh ¬¬()Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E}) is Boolean and Sh j()Sh ¬¬()Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E}), since it contains ext(j)ext(j), is dense in Sh j()Sh_j(\mathcal{E}) and there can be only one such dense Boolean subtopos. \qed

Proposition

The class Σ\Sigma of skeletal inclusions is the smallest class Γ\Gamma of geometric morphisms such that:

  • Γ\Gamma contains open inclusions and,

  • Γ\Gamma is closed under precomposition with dense inclusions: from gg dense, fΓf\in\Gamma and f,gf,g composable, follows fgΓfg\in\Gamma. \qed

The following exhibits the link between skeletal morphisms and Booleanness:

Proposition

A topos \mathcal{E} is Boolean iff all geometric morphisms \mathcal{F}\to\mathcal{E} to \mathcal{E} are skeletal.

Proof: When \mathcal{E} is Boolean it coincides with Sh ¬¬()Sh_{\neg\neg}(\mathcal{E}) hence ¬¬\neg\neg-sheaves of \mathcal{F} trivially have to land there.

Conversely, assume all \mathcal{F}\to\mathcal{E} are skeletal. By Barr's theorem, \mathcal{E} receives a surjective f:f:\mathcal{B}\to\mathcal{E} from a Boolean topos. ff being skeletal and surjective implies that im(f)=im(f)=\mathcal{E} is Boolean. \qed

A pullback characterisation of open geometric morphisms from Johnstone (2006, cor. 4.9):

Proposition

A geometric morphism f:f:\mathcal{F}\to\mathcal{E} is open iff the pullback of any bounded geometric morphism with codomain \mathcal{E} is skeletal. \qed

Remark

Skeletal morphisms between frames are studied in Banaschewski-Pultr (1994,1996), called weakly open there.

The equivalent concept for topological spaces appears in Mioduszewski-Rudolf (1969).

It is possible to define an analogous concept of m-skeletal geometric morphism using the De Morgan topology on a topos \mathcal{E} instead of ¬¬\neg\neg.

References

  • B. Banaschewski, A. Pultr, Variants of openness , Appl. Cat. Struc. 2 (1994) pp.1-21.

  • B. Banaschewski, A. Pultr, Booleanization , Cah. Top. Géom. Diff. Cat. XXXVII no.1 (1996) pp.41-60. (numdam)

  • Peter Johnstone, Factorization theorems for geometric morphisms II , pp.216-233 in LNM 915 Springer Heidelberg 1982.

  • Peter Johnstone, Sketches of an Elephant vol.II , Oxford UP 2002. (section D4.6, pp.1006-1010)

  • Peter Johnstone, Complemented sublocales and open maps , Annals of Pure and Applied Logic 137 (2006) pp.240–255.

  • Peter Johnstone, The Gleason Cover of a Realizability Topos , TAC 28 no.32 (2013) pp.1139-1152. (abstract)

  • J. Mioduszewski, L. Rudolf, H-closed and extremally disconnected Hausdorff spaces , Dissertationes Math. 66 1969. (toc)

Last revised on August 24, 2015 at 05:45:23. See the history of this page for a list of all contributions to it.