Abelian categories

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Requisitos previos obligatorios, hasta el fondo: 38

Aprenderás a

• Verify that R-Mod and the category of sheaves of abelian groups on a space are abelian, and explain why the category of free abelian groups is additive but not abelian.
• Prove the snake lemma and the five lemma in an abelian category.
• State the Freyd–Mitchell embedding theorem and explain what it does and does not allow.

Definición

• An additive category is a category enriched in abelian groups with a zero object and all finite biproducts.
• An abelian category is an additive category with all kernels and cokernels in which every monomorphism is a kernel and every epimorphism is a cokernel.
• A functor between abelian categories is exact if it preserves short exact sequences.

Teorema

• R-Mod is abelian for every ring R.
• Freyd–Mitchell: every small abelian category admits a full, faithful and exact functor into R-Mod for some ring R.
• Sheaves of abelian groups on a topological space form an abelian category with enough injectives (Grothendieck).
• In the category of free abelian groups, multiplication by 2 on ℤ is both monic and epic but not an isomorphism, so that category is not abelian.

Errores frecuentes

• Thinking every additive category is abelian.
• Applying Freyd–Mitchell to large categories without care; it is a theorem about small abelian categories.

Conduce a

Descubrimientos opcionales

• Sheaves Sheaves of abelian groups form an abelian category, the setting for sheaf cohomology.
• ← Homological algebra

Itinerarios de aprendizaje

• Research preparation

Lecturas complementarias

• Charles A. Weibel, An Introduction to Homological Algebra (1994)
• Peter Freyd, Abelian Categories (1964)
• Masaki Kashiwara and Pierre Schapira, Categories and Sheaves (2006)