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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
- A cocomplete locally small abelian category with a generator supplies the category-side SAFT hypotheses dually Corollary
- The pushout of a monomorphism is a monomorphism Corollary
- The opposite of abelian groups does not satisfy AB5 Counterexample
- Exactness at a node Definition
- FALSE under the Axiom of Choice: AB4 implies AB5 False statement
- Finite-diagonal cohomological double-complex spectral sequences Lemma
- Classical derived functors are the cohomology objects of the total derived functor Proposition
- Contravariant derived functors are derived on the opposite category Proposition
- Dual left-derived Grothendieck spectral sequence Remark
- The two halves of the four lemma are mutually dual Remark
- A bounded above acyclic complex of injective objects is contractible when its cycle monomorphisms split Theorem
- An additive functor is exact exactly when it preserves kernels and cokernels Theorem
- Cartan-Eilenberg injective resolutions exist Theorem
- Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel Theorem
- Exactness is self-dual Theorem
- Hom is left exact in each variable Theorem
- Injective comparison maps are unique up to cochain homotopy Theorem
- Injective comparison maps exist Theorem
- Injective object characterisations Theorem
- Kunneth Tor spectral sequence Theorem
- Left exactness, right exactness, and exactness are characterized by short exact sequences Theorem
- Nine lemma in an abelian category Theorem
- Short five lemma in an abelian category Theorem
- Symmetric nine lemma Theorem
- The acyclic-resolution theorem for right derived functors Theorem
- The diagram lemmas hold in the opposite category Theorem
- The dual Schanuel lemma for injective copresentations Theorem
- The horseshoe lemma for injective resolutions Theorem
- Universal coefficient spectral sequence 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)