nLab
simple object

Contents

Definition

Definition

An object XX in a category CC with a zero object 00 is simple if there are precisely two quotient objects of XX, namely 00 and XX.

Remark

If CC 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 XX is XX if and only if it is not 00.

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 CC, every morphism between simple objects is either a zero morphism or an isomorphism. If CC is also enriched in finite-dimensional vector spaces over an algebraically closed field, it follows that hom(X,Y)\hom(X, Y) has dimension 00 or 11.

Examples

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