Lattices of theories

In this chapter, by using the duality theorem established in Chapter 3, many ideas and concepts of elementary topos theory are transferred into the context of geometric logic; these notions notably include the coHeyting algebra structure on the lattice of subtoposes of a given topos, open, closed, quasi-closed subtoposes, the dense-closed factorization of a geometric inclusion, coherent subtoposes, subtoposes with enough points, the surjection-inclusion factorization of a geometric morphism, skeletal inclusions, atoms in the lattice of subtoposes of a given topos, the Booleanization and DeMorganization of a topos. An explicit description of the Heyting operation between Grothendieck topologies on a given category and of the Grothendieck topology generated by a given collection of sieves is also obtained, as well as a number of results about the problem of ‘relativizing’ a local operator with respect to a given subtopos.