Could not include topos theory - contents
In equivalent but extrinsic terms, a category is a localic topos if it is equivalent to the category of sheaves on a locale with respect to the topology of jointly epimorphic families (accordingly, every localic topos is a Grothendieck topos).
The frame of opens specifying the locale may indeed be taken as the poset of subobjects of (i.e., internal truth values). From the perspective of logic, localic toposes are those categories which are equivalent to the category of partial equivalence relations of the tripos given by a complete Heyting algebra (as before, the complete Heyting algebra may be taken as the poset of internal truth values).
Thus, in general we regard a localic geometric morphism as exhibiting E as a “localic S-topos”.
Here is the 2-category whose
objects are localic toposes over ;
The 2-category is actually a (1,2)-category; its 2-morphism are the pointwise ordering of frame homomorphisms. Thus this equivalence implies that is also a (1,2)-category, and moreover that it is locally essentially small, in the sense that its hom-categories are essentially small. (The 2-category of all toposes is not locally essentially small.) Assuming sufficient separation axioms, the hom-posets of , and hence , become discrete.
Many familiar toposes , even when they are not localic, can be covered by a localic slice (“covered” means the unique map is an epi). For example, if is a group, then is not itself localic, but it has a localic slice that covers it. Such a topos is called an etendue (see Lawvere’s 1975 monograph Variable Sets Etendu and Variable Structure in Topoi).1
A significant result due to Joyal and Tierney is that for any Grothendieck topos , there exists an open surjection where is localic. This fact is reproduced in Mac Lane and Moerdijk’s text Sheaves in Geometry and Logic (section IX.9), where the localic cover taken is the Diaconescu cover of .
Notice that a locale is itself a (Grothendieck) (0,1)-topos. Hence a localic topos is a 1-topos that behaves essentially like a (0,1)-topos. In the wider context this would be called a 1-localic (1,1)-topos.
Localic toposes are discussed around proposition 1.4.5 of section C.1.4 of