Cartesian monads on toposes Original Research Article

We consider two related questions: ‘When can a cartesian functor between toposes be factored as an inverse image functor followed by a direct image functor?’ and ‘Does every cartesian monad on a topos negated exists arise from a geometric morphism with codomain negated exists?’. The connection between the two questions is that the answer to the first, in the particular case when both the domain and codomain of the functor are the topos of sets, is ‘if and only if it carries a monad structure’. We investigate the class of cartesian monads on the topos of sets, providing a new proof that they correspond bijectively to strongly zero-dimensional locales in this topos. By combining two old results on strong functors, we show that any cartesian monad on an arbitrary topos has a canonical extension to an indexed monad; this suffices to extend the arguments employed in the topos of sets to an arbitrary Boolean base topos, but we also show how they fail in non-Boolean toposes. For such toposes, the answer to the second question therefore remains unknown.