A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of sheaves. But I was told one of the cool thing about topos is that sometimes there are morphisms of topos that are not from morphisms of a site. When people talk about this they mention the word "crystalline"...

But is there a toy example I can play around with? What's the easiest example of this?

elementarytopoi which are not equivalent to a category of sheaves on a site. – David Roberts Oct 31 '11 at 3:12