A geometric embedding is the right notion of embedding or inclusion of topoi $F \hookrightarrow E$, i.e. of subtoposes.
Notably the inclusion $Sh(S) \hookrightarrow PSh(S)$ of a category of sheaves into its presheaf topos or more generally the inclusion $Sh_j E \hookrightarrow E$ of sheaves in a topos $E$ into $E$ itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi.
Another perspective is that a geometric embedding $F \hookrightarrow E$ is the localizations of $E$ at the class $W$ or morphisms that the left adjoint $E \to F$ sends to isomorphisms in $F$.
The induced geometric morphism of a topological immersion $X \hookrightarrow Y$ is a geometric embedding. The converse holds if $Y$ is a $T_0$ space. (Example A4.2.12(c) in (Johnstone))
For $F$ and $E$ two topoi, a geometric morphism
is a geometric embedding if the following equivalent conditions are satisfied
the direct image functor $f_*$ is full and faithful (so that $F$ is a full subcategory of $E$);
the counit $\epsilon : f^* f_* \to Id_{F}$ of the adjunction $(f^* \dashv f_*)$ is an isomorphism
there is a Lawvere-Tierney topology on $E$ and an equivalence of categories $e : F \stackrel{\simeq}{\to} Sh_j E$ such that the diagram of geometric morphisms $\array{ F &\stackrel{f_*}{\to}& E \\ & {}_{e}\searrow^\simeq & \uparrow^{i} \\ && Sh_j E}$ commutes up to natural isomorphism $e^* i^* \simeq f^*$
That the first two conditions are equivalent is standard, that the third one is equivalent to the first two is for instance corollary 7 in section VII, 4 of (MacLaneMoerdijk)
There is a close relation between geometric embedding and localization: reflective localization.
Let $f : F \hookrightarrow E$ be a geometric embedding and let $W \subset Mor(E)$ be the class of morphisms sent by $f^*$ to isomorphisms in $F$.
We have:
$F$ is equivalent to the localization $E[W]^{-1}$;
$F$ is equivalent to the full subcategory of $E$ on $W$-local objects.
This fact connects for instance the description of sheafification in terms of geometric embedding $Sh(S) \hookrightarrow PSh(S)$ as described for instance in
with that in terms of localization at local isomorphisms, as described in
Moreover, this is the basis on which sheafification is generalized to (∞,1)-sheafification in
The following gives a detailed proof of the above assertion.
Write $\eta : Id_E \to f_* f^*$ for the unit of the adjunction.
Since $f_*$ is fully faithful we will identify objects and morphism of $F$ with their images in $E$. To further trim down the notation write $\bar {(-)} := f^*$ for the left adjoint.
Write $W$ for the class of morphism that are sent to isomorphism under $f^*$,
$E$ equipped with the class $W$ is a category with weak equivalences, in that $W$ satisfies 2-out-of-3.
Follows since isomorphisms satisfy 2-out-of-3.
$W$ is a left multiplicative system.
This follows using the fact that $f^*$ is left exact and hence preserves finite limits.
In more detail:
We have already seen in the previous proposition that
every isomorphism is in $W$;
$W$ is closed under composition.
It remains to check the following points:
Given any
with $w \in W$, we have to show that there is
with $w' \in W$.
To get this, take this to be the pullback diagram, $w' := h^* w$. Since $f^*$ preserves pullbacks, it follows that
is a pullback diagram in $F$ with $\bar w' = \bar h^* \bar w$. But by assumption $\bar w$ is an isomorphism. Therefore $\bar w'$ is an isomorphism, therefore $w'$ is in $W$.
Finally for every
with $w \in W$ such that the two composites coincide, we need to find
with $w' \in W$ such that the composites again coincide.
To get this, take $w'$ to be the equalizer of the two morphisms. Sending everything with $f^*$ to $F$ we find from
that $\bar r = \bar s$, since $\bar w$ is an isomorphism. This implies that $\bar w'$ is the equalizer
of two equal morphism, hence an identity. So $w'$ is in $W$.
For every object $a \in E$
the unit $\eta_a : a \to \bar a$ is in $W$;
if $a$ is already in $F$ then the unit is already an isomorphism.
This follows from the triangle identities of the adjoint functors.
and
In components they say that
for every $a \in E$ we have $(\bar a \stackrel{\bar \eta_a}{\to} \bar{\bar a} \stackrel{\simeq}{\to} \bar a) = Id_{\bar a}$
for every $a \in F$ we have $(a \stackrel{\eta_a}{\to} \bar a \stackrel{\simeq}{\to} a) = Id_a$
This implies the claim.
An object $a \in E$ is $W$-local object if for every $g : c \to d$ in $W$ the map
obtained by precomposition is an isomorphism.
Up to isomorphism, the $W$-local objects are precisely the objects of $F$ in $E$
First assume that $a \in F$. We need to show that $a$ is $W$-local.
Notice that the existence of the required isomorphism $Hom_F(d,a) \simeq Hom_F(c,a)$ is equivalent to the statement that for every diagram
there is a unique extension
To see the existence of this extension, hit the original diagram with $f^*$ to get
By the assumption that $c \to d$ is in $W$ the morphism $\bar c \to \bar d$ here is an isomorphism. By the assumption that $a$ is already in $F$ we have $\bar a \simeq a$ since the counit is an isomorphism. Therefore this diagram clearly has a unique extension
By the hom-isomorphism (using full faithfullness of $f_*$ to work entirely in $E$)
this defines a morphism $k : d \to a$. Chasing $k$ through the naturality diagram of the hom-isomorphism
shows that $k : d \to a$ does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.
So $a \in F$ is $W$-local.
Now for the converse, assume that a given $a$ is $W$-local.
By one of the above propositions we know that the unit $\eta_a : a \to \bar a$ is in $W$, so by the $W$-locality of $a$ it follows that
has an extension
By the 2-out-of-3 property of $W$ shown in one of the above propositions, (using that $Id_a$, being an isomorphism, is in $W$) it follows that $\rho_a : \bar a \to a$ is in $W$.
Since $\bar a$ is in $F$ and therefore $W$-local by the above, it follows that also
has an extension
So $\eta_a$ has a left inverse $\rho_a$ which itself has a left inverse $\lambda_a$. It follows that $\rho_a$ is also a right inverse to $\eta_a$, since
So if $a$ is $W$-local we find that $\eta_a : a \to \bar a$ is an isomorphism, hence that $a$ is isomorphic to an object of $F$.
$F$ is equivalent to the full subcategory $E_{W-loc}$ of $E$ on $W$-local objects.
By standard reasoning (e.g. KS lemma 1.3.11) there is a functor $F \to E_{W-loc}$ and a natural isomorphism
Since $F \hookrightarrow E$ and $E_{W-loc} \hookrightarrow E$ are full and faithful, so is $F \to E_{W-loc}$. Since by the above it is also essentially surjective, it establishes the equivalence $F \simeq E_{W-loc}$.
$F$ is equivalent to the localization $E[W^{-1}]$ of $E$ at $W$.
By one of the above propositions we know that $W$ is a left multiplicative systems.
This implies that the localization $E[W^{-1}]$ is (equivalent to) the category with the same objects as $E$, and with hom-sets given by
There is an obvious candidate for a functor
given on objects by the usual embedding by $f_*$ and on morphism by the map which regards a morphism trivially as a span with left leg the identity
For this to be an equivalence of categories we need to show that this is a essentially surjective and full and faithful functor.
To see essential surjectivity, let $a$ be any object in $E$ and let $\eta_a : a \to \bar a$ be the component of the unit of our adjunction on $a$, as above. By one of the above propositons, $\eta_a$ is in $W$. This means that the span
represents an element in $Hom_{E[W^{-1}]}(\bar a,a)$, and this element is clearly an isomorphism: the inverse is represented by
Since every $\bar a$ is in the image of our functor, this shows that it is essentially surjective.
To see fullness and faithfulness, let $a, b\in F$ be any two objects. By one of the above propositions this means in particular that $b$ is a $W$-local object. As discussed above, this means that every span
with $w \in W$ has a unique extension
But this implies that in the colimit that defines the hom-set of $E[W^{-1}]$ all these spans are identified with spans whose left leg is the identiy. And these are clearly in bijection with the morphisms in $Hom_E(a,b) \simeq Hom_F(a,b)$ so that indeed
for all $a,b \in F$. Hence our functor is also full and faithful and therefore define an equivalence of categories
There is a factorization system on the 2-category Topos whose left class is the surjective geometric morphisms and whose right class is the geometric embeddings. The factorization of a geometric morphism can be said to construct its image in the topos-theoretic sense.
See geometric surjection/embedding factorization.
Moreover, each geometric embedding itself has a (dense,closed)-factorization.
In the more general context of (∞,1)-topos theory an $(\infty,1)$-geometric embedding is an (∞,1)-geometric morphism
such that the right adjoint direct image $f_*$ is a full and faithful (∞,1)-functor.
See reflective sub-(∞,1)-category for more details.
Section VII, 4 of
and section A4.2 of