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The opposite of an abelian category is abelian
Statement
If is an abelian category, then the opposite category is also abelian.
Facts & Assumptions
Given: An abelian category .
An abelian category is additive and every morphism in it has a kernel and a cokernel (Abelian category).
The opposite of an additive category is additive (Additive categories are closed under passage to the opposite).
Passing to the opposite reverses every morphism while keeping the same objects (Opposite category ).
Proof
By [L1] and [L2], the opposite category is additive. Under [L3], a kernel in becomes a cokernel in , and a cokernel becomes a kernel, so every morphism of also has both.
The image of in the opposite category is the opposite of the coimage of , and the coimage of is the opposite of the image of . Therefore the canonical comparison for is the opposite of the canonical comparison for , which is an isomorphism by [L1]. So satisfies the same AB2 clause and is abelian.
Depends on
Used by
- The pushout of a monomorphism is a monomorphism Corollary
- An additive functor is exact exactly when it preserves kernels and cokernels Theorem
- Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel Theorem
- Left exactness, right exactness, and exactness are characterized by short exact sequences Theorem
Dependency tree · two levels
7 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 12.5.2 (standard reference, not scraped)