Simple objects

Last updated January 27, 2022

Definition

Let us work in a category with a zero object.

Definition. A simple object $A$ is one whose only quotient objects are $A$ and $0$.

Examples

1. In $\text{Vect}_k$, the only simple object up to isomorphism is $k$ itself.
   1. Any vector space of higher dimension can be mapped surjectively to one of lower non-zero dimension. $k$ can only be mapped surjectively to itself or to 0.