Category theory MOC

Abelian category

An abelian category 𝖠 is a preäbelian category such that every monomorphism is a kernel and every epimorphism is a cokernel. #m/def/cat Thus in particular an abelian category is enriched over 𝖠𝖻 and admits finite biproducts.

The prototypical example is 𝖠𝖻, or more generally 𝑅𝖬𝗈𝖽 for any ring 𝑅. The Freyd-Mitchell theorem gives a sense in which all abelian categories are categories of modules.

Properties

• Exact functor on abelian categories

Subtypes

• Spectral category
  • Semisimple category
    • Fusion category

#state/tidy | #lang/en | #SemBr