Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Abelian category
Definition
An abelian category is an additive category (Additive category) in which every morphism has a kernel and a cokernel, and in which for every morphism the canonical comparison morphism
constructed in The canonical morphism from the coimage to the image exists and is unique is an isomorphism.
The first two clauses are Grothendieck's AB1, while the invertibility of the canonical map is this page's working form of AB2.
Depends on
Used by
- An abelian category that is a preorder is trivial Corollary
- In an abelian category, monic means zero kernel and epic means zero cokernel Corollary
- An exact functor need not be faithful Counterexample
- Filtered vector spaces can be additive with kernels and cokernels without being abelian Counterexample
- Topological abelian groups are additive but not abelian Counterexample
- Torsion-free abelian groups do not form an abelian category Counterexample
- Abelian subcategory and exact embedding Definition
- Freyd-Mitchell gives a fully faithful exact functor from every small abelian category to a module category Remark
- The library does not use Freyd-Mitchell to prove the diagram lemmas Remark
- This page uses Grothendieck's AB1 and AB2 labels, and records the competing conventions Remark
- A left or right exact functor between abelian categories is automatically additive Theorem
- A pullback is the kernel of the difference of the two legs, and dually for pushouts Theorem
- A small product of abelian categories is abelian Theorem
- Abelian groups form an abelian category Theorem
- Additive functors from a small preadditive category to an abelian category form an abelian category Theorem
- An abelian category has all finite limits and all finite colimits Theorem
- An abelian category is balanced Theorem
- An equivalence between abelian categories is exact Theorem
- Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel Theorem
- Every morphism factors as an epimorphism followed by a monomorphism, uniquely up to unique isomorphism Theorem
- First isomorphism theorem in an abelian category Theorem
- Freyd and Mitchell's characterisation of abelian categories Theorem
- Freyd's axioms force the additive structure and recover the AB2 definition Theorem
- In a pullback square, the induced map on the kernels of the two parallel arrows is an isomorphism Theorem
- Left exactness, right exactness, and exactness are characterized by short exact sequences Theorem
- Modules over a ring form an abelian category Theorem
- The opposite of an abelian category is abelian Theorem
- The pullback of an epimorphism is an epimorphism Theorem
Dependency tree · two levels
9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Alexandre Grothendieck, Some aspects of homological algebra, §1.4 (standard reference, not scraped)
- The Stacks Project, Section 12.5, Definition 12.5.1 (standard reference, not scraped)
- Gautam Tamme, Algebra II Lecture 9, §9.4 (standard reference, not scraped)