This is a reverse of the question “Is there a finitely complete category with terminal object but NO subobject classifier?” From “An informal introduction to topos theory” by Tom Leinster I learned that there is 3 definitions of a subobject classifier in some category C:
- where we directly work with morphisms of C: for every monomorphism there exists a characteristic morphism etc.;
- a terminal object in the category of monomorphisms and pullback squares;
- the functor Sub is representable.
In order to make sense of (3) we need Sub which is defined via pullbacks in C. (3) requires C to have pullbacks. But (1) and (2) do not, though they imply existence of the terminal object. Is there a category with a subobject classifier and which is not finitely complete? (AFAIK subobject classifier → terminal object → (have pullbacks ↔ have finite limits = is finitely complete).)