Could not include topos theory - contents
In intrinsic terms, a topos is localic if it is generated under colimits by the subobjects of its terminal object $1$.
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 $1$ (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).
A Grothendieck topos $E$ is a localic topos if and only if its unique global section geometric morphism to Set is a localic geometric morphism.
Thus, in general we regard a localic geometric morphism $E \to S$ as exhibiting E as a “localic S-topos”.
Moreover, just localic topoi can be identified with locales, for any base topos $S$ the 2-category of localic $S$-topoi is equivalent to the 2-category Loc$(S)$ of internal locales in $S$.
Here $LocTopos(S)$ is the 2-category whose
objects are localic toposes over $S$;
morphisms are geometric morphisms, i.e. adjunctions in which the left adjoint preserves finite limits, considered as pointing in the direction of their right adjoint; and
2-morphisms are mate-pairs of natural transformations.
Then the 2-category $LocTopos$ is equivalent to the 2-category $Loc$ of locales (see C1.4.5 in the Elephant).
The 2-category $Loc$ is actually a (1,2)-category; its 2-morphism are the pointwise ordering of frame homomorphisms. Thus this equivalence implies that $LocTopos$ 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 $Topos$ of all toposes is not locally essentially small.) Assuming sufficient separation axioms, the hom-posets of $Loc$, and hence $LocTopos$, become discrete.
Obviously, every Grothendieck topos that is a category of sheaves on (the category of open subsets of) a topological space is localic.
Every sheaf topos over a posite is localic. (See there for details.)
Many familiar toposes $E$, even when they are not localic, can be covered by a localic slice $E/X$ (“covered” means the unique map $X \to 1$ is an epi). For example, if $G$ is a group, then $E = Set^G$ is not itself localic, but it has a localic slice $Set^G/G \simeq Set$ 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 $E$, there exists an open surjection $F \to E$ where $F$ 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 $E$.
In the context of (∞,1)-topos theory there is a notion of n-localic (∞,1)-topos.
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
The ‘etendu’ in the title of Lawvere’s monograph might not be a misspelled noun, but an adjective as part of a back translation of a (hypothetical) French expression ‘ensembles étendus’. See this nForum thread for some discussion and speculation on this point. ↩