Could not include topos theory - contents
A subtopos of a topos is a generalization of the concept of a subspace of a topological space.
The inclusions induce an ordering on the subtoposes of that makes them a lattice with a co-Heyting algebra structure i.e. the join operator has a left adjoint subtraction operator. This corresponds to a dual Heyting algebra structure on the Lawvere-Tierney topologies.
The atoms in the lattice of subtoposes of are precisely the Boolean two-valued subtoposes of (cf. Caramello (2009) prop.10.1 p.58).
(as every Grothendieck topos over ) is the classifying topos of some geometric theory and it can be shown that subtoposes of correspond precisely to deductively closed quotient theories of (cf. Caramello (2009), thm.3.6 p.15) i.e. passage to a subtopos corresponds to adding further geometric axioms to - localizing geometrically amounts to theory refinement logically.
F. Borceux, M. Korostenski, Open Localizations , JPAA 74 (1991) pp.229-238.
H. Forssell, Subgroupoids and quotient theories , TAC 28 no.18 (2013) pp.541-551. (pdf)