Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 22 / 4 / Solution

A localic geometric morphism $p:\mathcal{D}\to \mathcal{F}$ presents $\mathcal{D}$, over $\mathcal{F}$, as sheaves on an internal locale of $\mathcal{F}$. Equivalently, subobjects of inverse images $p^{\ast }X$ form a separating family. A hyperconnected geometric morphism $h$ has a full faithful inverse image $h^{\ast }$ whose essential image is closed under subobjects. In particular, it induces bijectionsFor this definition, subobject-closed means every subobject of $h^{\ast }X$ is isomorphic, over $h^{\ast }X$, to the inverse image of a subobject of $X$.

Here is the construction and proof of the hyperconnected-localic factorization. Given $f:\mathcal{E}\to \mathcal{F}$, its right adjoint gives natural bijectionsLet $L$ be the internal locale whose frame of opens is $f_{\ast }{\Omega }_{\mathcal{E}}$. The frame operations are determined by these bijections: intersections are intersections of represented subobjects; the join of a family is its image under the projection from the parameter-indexed family. For an internal index object $I$, evaluation of the family over $X\times I$ becomes a subobject of $f^{\ast }X\times f^{\ast }I$, and its direct image to $f^{\ast }X$ represents the join. Pullback stability of images makes this natural in $X$, and distributivity follows from the corresponding subobject identity in $\mathcal{E}$. This explicitly constructs the internal frame rather than asserting it from the existence of $f_{\ast }$ alone.

Set $\mathcal{D}={\mathrm{Sh}}_{\mathcal{F}}(L)$, with its localic projection $p$ to $\mathcal{F}$. We construct $h^{\ast }:\mathcal{D}\to \mathcal{E}$. A basic open family is a map $u:X\to f_{\ast }{\Omega }_{\mathcal{E}}$; its represented object in $\mathcal{E}$ is the subobject $A_u\hookrightarrow f^{\ast }X$ corresponding to $u$. Meets and covering joins of open families become intersections and jointly epic families of these subobjects.

An internal sheaf on $L$ is presented by its local sections with their open domains and restrictions. Pull these domains back to the corresponding $A_u$ and identify two sections wherever their restrictions agree. This is an effective equivalence relation in $\mathcal{E}$, so its quotient defines $h^{\ast }$ of the sheaf. More concretely, if $S$ is the object of sections and $d:S\to \mathcal{O}(L)$ gives their domains, start with $A_d\subseteq f^{\ast }S$. For two sections, the join of the opens on which they agree gives an $L$-valued relation on $S\times S$, hence a subobject of $f^{\ast }(S\times S)$. Restricted to $A_d\times A_d$, this is the germ equivalence relation used for the quotient. The restriction and sheaf axioms guarantee reflexivity, symmetry and transitivity. This construction is functorial. Products and equalizers are obtained by meeting domains and equalizing restrictions, so it preserves finite limits. A right adjoint is obtained by taking, over each open family $u$, the parameterized maps $A_u\to Y$ for $Y\in \mathcal{E}$. These maps satisfy the sheaf condition because a covering join gives an effective gluing of the $A_u$; their parameter objects are supplied by the original adjunction $f^{\ast }⊣f_{\ast }$ and exponentials. The map bijection on basic open families, followed by gluing, proves this is right adjoint to $h^{\ast }$. Thus $h$ is a geometric morphism.

We must check hyperconnectedness, not just construct a factorization. A map $A_u\to A_v$ has a graph which is a subobject of $f^{\ast }(X\times Y)$. The represented-subobject bijection turns it into an $L$-valued relation on $X\times Y$. Its totality and single-valuedness say exactly that this relation is a map between the corresponding basic sheaf objects: totality uses the covering join, and single-valuedness uses intersections and the diagonal of $Y$. Thus maps between basic open families are precisely their maps in $\mathcal{E}$. Applying this to the local sections of arbitrary sheaves, compatible maps glue uniquely on both sides. Hence $h^{\ast }$ is full and faithful. Likewise a subobject of $h^{\ast }S$ pulls back to subobjects of its basic $A_u$; each is again represented by an open family, and their compatibility glues to a subsheaf of $S$. The essential image is therefore subobject-closed, proving that $h$ is hyperconnected.

The inverse image $p^{\ast }X$ is the constant family over the whole locale. The construction sends it to $f^{\ast }X$, so $h^{\ast }{p}^{\ast }\cong f^{\ast }$; taking right adjoints gives the required factorizationThis argument uses the usual effective quotient and local-section constructions in a topos; it also explains why the relevant internal opens are relative subobjects of $f^{\ast }X$, not just global subobjects of the terminal object.

For uniqueness, suppose $f=p^{\prime }{h}^{\prime }$ with $h^{\prime }$ hyperconnected and $p^{\prime }$ localic. Then for every $X$,If ${\mathcal{D}}^{\prime }$ is sheaves on an internal locale $L^{\prime }$, the right side is represented by its frame $\mathcal{O}(L^{\prime })$. These identifications preserve intersections, joins and parameter change, so the representing internal frames are isomorphic: $\mathcal{O}(L^{\prime })\cong f_{\ast }{\Omega }_{\mathcal{E}}$. The locale and its sheaf topos are therefore equivalent over $\mathcal{F}$. Under this equivalence the basic open families have the same images in $\mathcal{E}$, and their local-section presentations determine all other images and maps. Thus the two hyperconnected parts agree up to the same equivalence. The factorization is unique up to equivalence.

Finally suppose $f=hi$ with $i:\mathcal{E}\hookrightarrow \mathcal{D}$ a geometric embedding and $h:\mathcal{D}\to \mathcal{F}$ hyperconnected. Let $j$ be the local operator on $\mathcal{D}$ presenting the inclusion. For a subobject $A\hookrightarrow X$ of $\mathcal{F}$, take the $j$-closure of $h^{\ast }A$ in $h^{\ast }X$. Hyperconnectedness makes it the inverse image of a unique subobject of $X$. This defines a closure operation on $\mathcal{F}$; its extensivity, idempotence, intersection preservation and pullback stability descend from the $j$-closure. Let $k$ be its local operator.

Hyperconnectedness also gives $h_{\ast }{\Omega }_{\mathcal{D}}\cong {\Omega }_{\mathcal{F}}$, by the represented-subobject bijections. The direct image $i_{\ast }{\Omega }_{\mathcal{E}}$ is the closed-truth object ${\Omega }_j$, and henceThis follows either by taking the equalizer of $j$ and the identity or by representing closed subobjects in the preceding closure construction. The internal locale with frame ${\Omega }_k$ is the sublocale of the terminal locale determined by $k$: covering joins are exactly closure by $k$. Its sheaves are the $k$-sheaves in $\mathcal{F}$, and its projection is their geometric inclusion. Applying the just-proved factorization to $f$ therefore givesThis supplies the requested reordered factorization.