nLab skeletal geometric morphism

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

• $f$ restricts to a geometric morphism $Sh_{\neg\neg}(\mathcal{F}) \to Sh_{\neg\neg}(\mathcal{E})$ .

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

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

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 subtoposes$i:Sh_j(\mathcal{E})\hookrightarrow \mathcal{E}$ are precisely those subtoposes with $Sh_{\neg\neg}(Sh_j(\mathcal{E}))=Sh_{\neg\neg}(\mathcal{E})$ (cf. this proposition) and, therefore, are skeletal.

Properties

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

Proposition

An inclusion $i:Sh_j(\mathcal{E})\hookrightarrow \mathcal{E}$ is skeletal iff $ext(j)$ , the $j$-closure of $0\rightarrowtail 1$ , is a $\neg\neg$-closed subterminal object of $\mathcal{E}$.

Proof: First, notice that in general for a topology $j$ a subterminal $U$ is $j$-closed iff it is a $j$-sheaf since $1$ is always a $j$-sheaf. Hence $0$ is $\neg\neg$-closed precisely when it is a $\neg\neg$-sheaf.

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

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

Proposition

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

• $\Gamma$ contains open inclusions and,

• $\Gamma$ is closed under precomposition with dense inclusions: from $g$ dense, $f\in\Gamma$ and $f,g$ composable, follows $fg\in\Gamma$. $\qed$

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 $Sh_{\neg\neg}(\mathcal{E})$ 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 $f:\mathcal{B}\to\mathcal{E}$ from a Boolean topos. $f$ being skeletal and surjective implies that $im(f)=\mathcal{E}$ is Boolean. $\qed$

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

Proposition

A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is open iff its pullback along 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 $\neg\neg$.

Related entries

• open geometric morphism

• dominant geometric morphism

• proper geometric morphism

• double negation

• De Morganization

• De Morgan topos

References

• B. Banaschewski, A. Pultr, Variants of openness, Appl. Cat. Struc. 2 (1994) 1-21 [doi:10.1007/BF00873038]

• B. Banaschewski, A. Pultr, Booleanization, Cah. Top. Géom. Diff. Cat. XXXVII 1 (1996) 41-60 [numdam:CTGDC_1996__37_1_41_0]

• 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)