category theory

# Contents

## 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

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.

Revised on September 7, 2014 05:30:18 by Anonymous Coward (90.6.35.180)