The concept of an open subtopos generalizes the concept of an open subspace from topology to toposes.
Let $U$ be a subterminal object of a topos. Then $o_U(V)\coloneqq (U\Rightarrow V)$ defines a Lawvere-Tierney topology on $\mathcal{E}$, whose corresponding subtopos is called the open subtopos associated to $U$.
The reflector into the topos of sheaves can be constructed explicitly as $O_U(X) = X^U$.
A topology that is of this form for some subterminal object $U$ is called open.
In case $\mathcal{E}=Sh(X)$ is the topos of sheaves on a topological space $X$, a subterminal object is just an open subset $U$ of $X$ and the open subtopos corresponding to it is equivalent to $Sh(U)$.
As one would expect from the topological situation, for any topos $\mathcal{E}$, the empty subtopos (given by $o(V) \coloneqq (\bot \Rightarrow V) = \top$) and $\mathcal{E}$ itself (given by $o(V) \coloneqq (\top \Rightarrow V) = V$) are open subtoposes of $\mathcal{E}$.
The subterminal object $U$ in $\mathcal{E}$ is associated with a closed subtopos $Sh_{c(U)}(\mathcal{E})$ as well e.g. in the case of $\mathcal{E}=Sh(X)$ on a space $X$ this yields $Sh(X\setminus U)$.
Moreover, given a Lawvere-Tierney topology $j$ on a topos $\mathcal{E}$ with corresponding subtopos $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$, we a get a canonical subterminal object $ext(j)$ associated to $j$ by taking the $j$-closure of $O\rightarrowtail 1$. The corresponding closed and open subtoposes associated to $ext(j)$ provide a ‘closure’ $Sh_{c(ext(j))}(\mathcal{E})$, respectively, an ‘exterior’ $Sh_{o(ext(j))}(\mathcal{E})$ for $Sh_j(\mathcal{E})$ (cf. SGA4, p.461). More on this below.
Let $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ be a subtopos with corresponding topology $j$. The following are equivalent:
See Johnstone (1980, pp.219-220; 2002, pp.609-610).
Let $U$ a subterminal object and $Sh_{c(U)}(\mathcal{E})$ and $Sh_{o(U)}(\mathcal{E})$ the corresponding closed, resp. open subtoposes. Then $Sh_{c(U)}(\mathcal{E})$ and $Sh_{o(U)}(\mathcal{E})$ are complements for each other in the lattice of subtoposes.
See Johnstone (2002, pp.212,215).
Whereas, general open morphisms are only bound to preserve first order logic, open inclusions preserve also higher order logic since their inverse images are logical.
That the inverse image is logical is a special case of the general fact that the pullback functor $\mathcal{E}\to\mathcal{E}/X$ along $X\to 1$ is logical for arbitrary objects $X$. In particular, $Sh_{o(U)}(\mathcal{E})\cong\mathcal{E}/U$.
Subtoposes of a topos $\mathcal{E}$ correspond to localizations of $\mathcal{E}$ i.e. replete, reflective subcategories whose reflector preserves finite limits. Just as this notion makes sense more generally for categories with finite limits, the notion of open localization makes sense more generally for locally presentable categories:
Given a locally $\alpha$-presentable category $\mathcal{C}$ with subcategory of $\alpha$-presentable objects $\mathcal{P}$, the subobject $\Omega_\mathcal{C}$ of the subobject classifier of $Set^{\mathcal{P}^{op}}$ given by $\Omega_\mathcal{C}(P):=$ set of $\alpha$-exact subpresheaves of $\mathcal{P}(\text{_},P)$, classifies subobjects of $\mathcal{C}$. Furthermore, localizations of $\mathcal{C}$ correspond to topologies $j:\Omega_\mathcal{C}\to\Omega_\mathcal{C}$.
At this level of generality, an open subtopos of a Grothendieck topos corresponds to the notion of an open localization of a locally presentable category that is studied in Borceux-Korotenski (1991). The main result in their paper is the following
Let $l\dashv i: \mathcal{D}\hookrightarrow\mathcal{C}$ be a localization of a locally presentable category $\mathcal{C}$. The localization is called open if the following equivalent conditions are satisfied:
The corresponding closure operator admits a universal dense interior operator.
The associated topology $j_\mathcal{D}:\Omega_\mathcal{C}\to\Omega_\mathcal{C}$ has a left adjoint.
The localization is essential with $k\dashv l$ and, additionally, the first of the following diagrams^{1} being a pullback implies the second being a pullback, too:
In case $\mathcal{C}$ is a topos, $i:\mathcal{D}\hookrightarrow\mathcal{C}$ is an open subtopos.
Open localizations are special cases of essential localizations, and are in general better behaved than the latter. For example, the meet of two essential localizations in the lattice of essential localizations does not coincide with their meet in the lattice of localizations. Compare this with the following
Let $\mathcal{C}$ a locally presentable category. The meet of two open localizations in the lattice of localizations is an open localization.
cf. Borceux-Korotenski (1991, p.235).
Let $\mathcal{C}$ a locally presentable category where unions are universal. The supremum of a family of open localizations in the lattice of all localizations is again an open localization.
cf. Borceux-Korotenski (1991, p.236).
This applies e.g. to Grothendieck toposes since they are locally presentable and colimits are universal.
Let $\mathcal{C}$ a locally presentable category where unions are universal. The open localizations in $\mathcal{C}$ constitute a locale.
cf. Borceux-Korotenski (1991, p.237).
The following proposition closes the circle and recovers the primordial example of sheaf subtoposes $Sh(U)$ on open subsets $U$ as a special case:
Let $\mathcal{C}$ a locally presentable category in which colimits are universal. Then the locale of open localizations of $\mathcal{C}$ is isomorphic to the locale of subterminal objects of $\mathcal{C}$.
cf. Borceux-Korotenski (1991, p.238).
Let $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ be a subtopos of a (Grothendieck) topos with corresponding topology $j$. From prop. it follows that the supremum of the family of all open subtoposes contained in $Sh_j(\mathcal{E})$ is open again and, since it coincides with the supremum in the lattice of all localizations, is contained in $Sh_j(\mathcal{E})$. Clearly, it is the biggest open subtopos contained in $Sh_j(\mathcal{E})$ and therefore called the interior of $Sh_j(\mathcal{E})$, denoted by $Sh_{o(int(j))}(\mathcal{E})$ and the corresponding subterminal object by $int(j)$.
Whereas the other open subtopos $Sh_{o(ext(j))}(\mathcal{E})$ connected with $Sh_j(\mathcal{E})$ corresponding to $ext(j)$ is the biggest open subtopos disjoint from $Sh_j(\mathcal{E})$ i.e. its exterior. Then the sum $Sh_{o(ext(j))}(\mathcal{E}) \vee Sh_{o(int(j))}(\mathcal{E})$ is open again and corresponds to the subterminal object $ext(j)\vee int(j)$. Its closed complement $Sh_{c(ext(j)\vee int(j))}(\mathcal{E})$ is called the boundary of $Sh_j(\mathcal{E})$ in (SGA 4, p. 461).
For some further details see at dense subtopos.
M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (Exposé IV 9.2, 9.3.4-9.4., pp.451ff)
F. Borceux, M. Korostenski, Open localizations , JPAA 74 (1991) 229-238 doi
C. Getz, M. Korostenski, Open localizations and factorization systems , Quest. Math. 17 no.2 (1994) 225-230 doi
Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014, 93-95)
Peter Johnstone, Open maps of toposes , Manuscripta Math. 31 (1980) 217-247. (gdz)
Peter Johnstone, Sketches of an Elephant vols. I,II, Oxford UP 2002. (A4.5., pp.204-220; C3.1.5-7, pp.609f)
Here $f$ corresponds to $\bar{f}$, resp. $g$ to $\bar{g}$, under the adjunction $k\dashv l$. ↩