An object in a category with a zero object is simple if there are precisely two quotient objects of : and . If is abelian, we may use subobjects in place of quotient objects in the definition, and this is more common; the result is the same.
Note that itself is not simple, as it has only one quotient object. It is too simple to be simple.
In constructive mathematics, we want to phrase the definition as: a quotient object of is if and only if it is not .
In an abelian category , every morphism between simple objects is either a zero morphism or an isomorphism. If is also enriched in finite-dimensional vector spaces over an algebraically closed field, it follows that has dimension or .
A simple Lie algebra is a simple object in LieAlg that also is not abelian. As an abelian Lie algebra is simply a vector space, the only simple object of that is not accepted as a simple Lie algebra is the -dimensional Lie algebra.