Simple object - Knowlpedia

A simple object of an abelian category $\mathcal A$ is an object $S\not\cong0$ such that every monomorphism $U\hookrightarrow S$ has either $U\cong0$ or is an isomorphism. Equivalently, its only subobjects are $0$ and $S$ , where subobjects are monomorphisms into $S$ identified up to isomorphism over $S$ .

Examples and a distinction

In vector spaces, the simple objects are the one-dimensional spaces. In a module category they are precisely simple modules. All one-dimensional $k$ -vector spaces are isomorphic, so there is one isomorphism class of simples, although there are many objects.

A simple object cannot split as a direct sum of two nonzero objects. The converse fails: the abelian group $\mathbb Z/4\mathbb Z$ has the proper nonzero subgroup $2\mathbb Z/4\mathbb Z$ , but cannot be written as a direct sum of two nonzero subgroups.

Endomorphisms

A nonzero map between simple objects is an isomorphism: its kernel must vanish and its image must be the target. Thus $\operatorname{End}(S)$ is a division ring. If the category is $k$ -linear, Hom spaces are finite-dimensional, and $k$ is algebraically closed, then $\operatorname{End}(S)=k\operatorname{id}_S$ . Indeed, every endomorphism has an eigenvalue in the sense that its minimal polynomial over $k$ has a root; in a division algebra this forces that endomorphism to be scalar.

References

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), §1.5 and Proposition 1.8.4; author-hosted text.