(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
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.