Minimal objects in category theory The notion of initial object in category theory, corresponds to the notion of minimum element in a pre-ordered set. Is there a notion of minimal object in category theory (that would correspond to the notion of minimal element in a pre-ordered set)?
 A: One categorical way of characterizing minimal objects of a poset is as an $x$ such that if there's a morphism $y\to x$ then it's an isomorphism. This isn't a particularly common phenomenon in category theory, except for the case of strict initial objects, an initial object which is also minimal in this sense. Note that initial objects in a category don't have to be minimal, since even if some $x$ maps into an initial object $0,$ not all maps $x\to y$ must factor through $0$ (unlike in the case of a poset.)
There is also a generalization of a poset with enough minimal objects (where every object is comparable to some minimal, as is important for instance in the opposite of the poset of ideals of a ring.) This is the idea of a category with a weakly initial set, a set of objects such that each object of the category admits a map from some object in the set. These sets are very important in adjoint functor theorems, although again, such objects don't need to be minimal in any particular sense in general.
