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, and the endomorphism algebra of any simple object is a division ring.

If $C$ is also enriched in finite-dimensional vector spaces over an algebraically closed field $k$, then $hom(X, Y)$ has dimension $0$ or $1$ for any pair of simple objects $X$ and $Y$. In this case the endomorphism algebra of any simple object is $k$.

Proof

Suppose $X$ and $Y$ are simple objects in an abelian categor $C$. If $f \colon X \to Y$ is any morphism then the kernel of $f$ must be either $0$ or $X$, while its kernel must be $0$ or $Y$. If the kernel and cokernel are both $0$, $f$ is an isomorphism; otherwise $f = 0$. It follows that every element of the endomorphism ring $End(X)$ of a simple object $X$ is zero or invertible, so $End(X)$ is a division ring.

Next suppose $C$ is enriched over finite-dimensional vector spaces over an algebraically closed field $k$. In this case $End(X)$ is a finite-dimensional division algebra over $k$, but any such algebra is isomorphic to $k$. Post-composing with an isomorphism $f: X \to Y$ gives a vector space isomorphism $End(X) \to hom(X,Y)$, so if such an isomorphism $f$ exists then $hom(X,Y)$ is one-dimensional. If no such isomorphism exists all the morphisms from $X$ to $Y$ are zero.

Remark

If an abelian category is enriched over finite-dimensional vector spaces over a field $k$ that is not algebraically closed, the endomorphism algebra of a simple object can be a division algebra other than $k$. For example consider $k = \mathbb{R}$. In the category of real representations of the Lie group SO(2), the usual action of rotations on the plane gives a simple object (that is, irreducible representation) $X$ with $End(X) \cong \mathbb{C}$. In the category of real representations of Sp(1))?, the action of this group by right multiplication on the quaternions gives a simple object $X$ with $End(X) \cong \mathbb{H}$.

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