Cohomology and homotopy
In higher category theory
A subtopos of a topos is a generalization of the concept of a subspace of a topological space.
For a topos, a subtopos is another topos equipped with a geometric embedding .
If this is an open geometric morphism (or an essential geometric morphism) one speaks of an open subtopos (an essential subtopos, respectively, also called a level of a topos).
Sheaves, localization, closure and reflection
If is an elementary topos then subtoposes correspond to Lawvere-Tierney topologies on , to localizations of as well as to universal closure operators on .
For classifying toposes
Every Grothendieck topos over is (equivalent to) the classifying topos of some geometric theory and it can be shown that subtoposes of correspond precisely to deductively closed quotient theories of (Caramello (2009); thm. 3.6) i.e. passage to a subtopos corresponds to adding further geometric axioms to - localizing geometrically amounts to theory refinement logically.
The lattice of subtoposes
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 lattice structure is further analyzed in (Johnstone (2002), Caramello (2009)). We contend us to report
Let , be two subtoposes of a topos and the corresponding topologies with their join in the lattice of topologies. Then the following holds: .
(cf. Johnstone (2002), p.217). In other words, the meet of two subtoposes is just their intersection (and is equivalently given by the subtopos corresponding to the join of their topologies).
The atoms in the lattice of subtoposes of are precisely the two-valued Boolean subtoposes of .
(Caramello (2009); prop. 10.1). This follows from the fact that two-valued and Boolean toposes are opposite extremes when it comes to dense subtoposes: in a two-valued topos every non-trivial subtopos is dense, whereas a Boolean topos has no non-trivial dense subtopos (cf. at dense subtopos for further details).
M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (exposé IV, section 9, pp.431ff)
F. Borceux, M. Korostenski, Open Localizations , JPAA 74 (1991) pp.229-238.
O. Caramello, pp.15,58 of Lattices of theories, (2009). (arXiv:0905.0299)
H. Forssell, Subgroupoids and quotient theories , TAC 28 no.18 (2013) pp.541-551. (pdf)
Peter Johnstone, Sketches of an Elephant I, Oxford UP 2002. (pp.195-223)
Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014, section 3.5)
G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations , Bull. Soc. Math. de Belgique XLI (1989) pp.261-299.