nLab simple object

Definition

Definition

An object $X$ in a category $C$ with a zero object $0$ is simple if there are precisely two quotient objects of $X$, namely $0$ and $X$.

Remark

If $C$ is abelian, we may use subobjects in place of quotient objects in the definition, and this is more common; the result is the same.

Remark

The zero object itself is not simple, as it has only one quotient object. It is too simple to be simple.

Remark

In constructive mathematics, we want to phrase the definition as: a quotient object of $X$ is $X$ if and only if it is not $0$.

Definition

An object which is a direct sum of simple objects is called a semisimple object.

Properties

In an abelian category

Proposition

(Schur's lemma)

In an abelian category $C$, every morphism between simple objects is either a zero morphism or an isomorphism.

If $C$ is also enriched in finite-dimensional vector spaces over an algebraically closed field, it follows that $\hom(X, Y)$ has dimension $0$ or $1$.

Examples

• In the category Vect of vector spaces over some field $k$, the simple objects are precisely the lines: the $1$-dimensional vector spaces, i.e. $k$ itself, up to isomorphism.

• A simple group is a simple object in Grp. (Here it is important to use quotient objects instead of subobjects.)

• For $G$ a group and $Rep(G)$ its category of representations, the simple objects are the irreducible representations.

• A simple ring is not a simple object in Ring (which doesn't have a zero object anyway); instead it is a ring $R$ that is simple in its category of bimodules.

• 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 $Lie Alg$ that is not accepted as a simple Lie algebra is the $1$-dimensional Lie algebra.

Related entries

• length of an object

• semisimple category

  • fusion category

  • modular tensor category