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In an abelian category, monic means zero kernel and epic means zero cokernel
Statement
For a morphism in an abelian category:
- is monic if and only if its kernel is zero;
- is epic if and only if its cokernel is zero.
Facts & Assumptions
Given: An abelian category and a morphism in it.
An abelian category is additive, hence preadditive and equipped with a zero object (Abelian category).
In a preadditive category with a zero object, a morphism is monic exactly when its kernel is zero (In a preadditive category with a zero object, a morphism is monic exactly when its kernel is zero).
In a preadditive category with a zero object, a morphism is epic exactly when its cokernel is zero (In a preadditive category with a zero object, a morphism is epic exactly when its cokernel is zero).
Proof
The monomorphism claim is exactly [L2], because [L1] supplies the preadditive and zero-object hypotheses that [L2] needs.
The epimorphism claim is exactly [L3], for the same reason.
Depends on
Used by
- FALSE: every infinite coproduct of complexes has homology equal to the coproduct of their homologies False statement
- Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
- Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
- Injective abelian sheaves are flasque Lemma
- The Hom functor of a small projective generator is exact, coproduct-preserving, and faithful Lemma
- A subcomplex is the kernel of its quotient map Proposition
- A chain map is a quasi-isomorphism exactly when its cone is acyclic Theorem
- A complex is exact at n exactly when its nth homology is zero Theorem
- A left exact functor preserves monomorphisms and a right exact functor preserves epimorphisms Theorem
- A short exact sequence is a kernel-cokernel pair Theorem
- Degenerate exactness criteria Theorem
- Finite abelian categories admit finite-dimensional module models Theorem
- The pullback of an epimorphism is an epimorphism Theorem
Dependency tree · two levels
11 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
- The Stacks Project, Section 12.5, Lemma 12.5.4 (standard reference, not scraped)