  The classifying topos of a geometric theory is a topos such that geometric
morphisms into it correspond to models of that theory. We study classifying
toposes for different infinitary logics: first-order, sub-first-order (i.e.
geometric logic plus implication) and classical. For the first two, the
corresponding notion of classifying topos is given by restricting the use of
geometric morphisms to open and sub-open geometric morphisms respectively. For
the last one, we restrict ourselves to Boolean toposes instead. Butz and
Johnstone proved that the first-order classifying topos of an infinitary
first-order theory exists precisely when that theory does not have too many
infinitary first-order formulas, up to intuitionistically provable equivalence.
We prove similar statements for the sub-first-order and Boolean case. Along the
way we obtain completeness results of infinitary sub-first-order logic and
infinitary classical logic with respect to (Boolean) toposes.
