Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27‡ rests on unproved material (inherited)
How 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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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 C in which every morphism has a kernel and a cokernel.

[L1]

An additive category has finite biproducts and is preadditive (Additive category).

[L2]

In a preadditive category, finite products are biproducts (In a preadditive category, a finite product is automatically a biproduct).

[L3]

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).

[L4]

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

technique · direct
1.1

By [L1], the category already has finite biproducts, hence finite products and finite coproducts.

L1L2
1.2

By hypothesis every morphism has a kernel and a cokernel. Therefore [L3] gives an equalizer and a coequalizer for every parallel pair.

L3
2.1

Combining steps 1.1 and 1.2 with the finite-(co)limit criteria of [L4] yields all finite limits and all finite colimits.

L4step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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