nLab
simple object

Definition

An object $X$ in a category $C$ with a zero object $0$ is simple if there are precisely two subobjects of $X$: $0$ and $X$.

Note that $0$ itself is not simple, as it has only one subobject. This is similar to the issues discussed at connected space.

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

Examples

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

• For $G$ a group and $\mathrm{Rep}(G)$ 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 a simple in its category of bimodules.

Related entries

• semisimple category

  • fusion category

  • modular tensor category