There are multiple conceivable such generalizations, depending in particular on whether one tries to generalize the notion of Grothendieck topos or of elementary topos, and in the latter case what axioms one chooses to take as the basis for generalization.
In contrast, (2,1)-toposes are much better understood.
See also higher topos theory.
The 2-toposes of 2-sheaves over a 2-site are special among all 2-toposes, in direct generalization of how sheaf toposes (“Grothendieck toposes”) are special among all toposes. In that case, Giraud's theorem famously characterizes sheaf toposes. This characterization has a 2-categorical analog: the 2-Giraud theorem.
In particular a -localic 2-topos is the same as a (2,1)-topos.
For a 2-topos, there is an equivalence of 2-categories
If is -localic, with a (2,1)-site of definition , then there is already an equivalence
with the 2-category of categories internal to the underlying (2,1)-topos.
If is -localic, with 1-site of definition, then there is even already an equivalence
with the internal categories in the underlying sheaf topos.
By the discussion at n-localic 2-topos, a 2-sheaf 2-topos has enough groupoids precisely if it has a (2,1)-site of definition, and has enough discretes precisely if it has a 1-site of definition. With this the second and third statement is this theorem at 2-congruence.
The noteworthy point about theorem 1 is that for an ambient context which is a -localic (2,1)-topos, the straightforward morphisms of internal categories, hence the notion of internal functors, needs no further localization. This is in stark contrast to the situation for an ambient 1-category. The generalization of this phenomenon is discussed at category object in an (∞,1)-category.
An introduction is in
Early developments include
A detailed discussion from the point of view of internal logic is at
Discussion of the elementary topos-analog of 2-toposes is in