Category theory MOC

Simple object

Let 𝖢 be a category. An object 𝑋 ∈𝖢 is called simple iff its only quotients (in the sense of coëqualizers) are the terminal object and 𝑋 itself.¹ cat If 𝖢 is abelian category, it is sufficient for 𝑋 to have no subobjects.

Examples

• Simple group in Category of groups.
• Simple module in Category of left modules.
• Simple ring in Category of rings.

Properties

• Schur’s lemma in abelian categories

See also

• Semisimple object in an abelian category

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Footnotes

1. Constructively, a quotient 𝑌 of 𝑋 is 𝑋 iff it is not 𝟙. ↩