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An additive category with all kernels and cokernels has all finite limits and colimits
Statement
If an additive category has a kernel and a cokernel for every morphism, then it has all finite limits and all finite colimits.
Facts & Assumptions
Given: An additive category in which every morphism has a kernel and a cokernel.
An additive category has finite biproducts and is preadditive (Additive category).
In a preadditive category, finite products are biproducts (In a preadditive category, a finite product is automatically a biproduct).
Equalizers are kernels of differences, and coequalizers are cokernels of differences (In a preadditive category, the equalizer of a parallel pair is the kernel of their difference, In a preadditive category, the coequalizer of a parallel pair is the cokernel of their difference).
Finite limits are equivalent to finite products and equalizers, and finite colimits to finite coproducts and coequalizers (Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals).
Proof
By [L1], the category already has finite biproducts, hence finite products and finite coproducts.
By hypothesis every morphism has a kernel and a cokernel. Therefore [L3] gives an equalizer and a coequalizer for every parallel pair.
Combining steps 1.1 and 1.2 with the finite-(co)limit criteria of [L4] yields all finite limits and all finite colimits.
Depends on
- Additive category
- In a preadditive category, a finite product is automatically a biproduct
- In a preadditive category, the equalizer of a parallel pair is the kernel of their difference
- In a preadditive category, the coequalizer of a parallel pair is the cokernel of their difference
- Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals
Used by
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Sources
- The Stacks Project, Section 12.3: Preadditive and additive categories (standard reference, not scraped)