Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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  • 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.

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An equivalence between abelian categories is exact

Statement

Every equivalence between abelian categories is exact.

Facts & Assumptions

Given: An equivalence F:A→B between abelian categories.

[L1]

Equivalences preserve all limits and colimits that exist (Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense).

[L2]

Abelian categories are additive (Abelian category).

[L3]

A functor between additive categories is additive exactly when it preserves finite biproducts (A functor between additive categories is additive exactly when it preserves finite biproducts).

[L4]

Exact means additive, left exact, and right exact (Left exact and right exact functors, Exact functor between abelian categories).

Proof

technique · direct
1.1L1L4

By [L1], the functor F preserves finite limits and finite colimits. So by [L4] it is left exact and right exact. In particular it preserves finite products and finite coproducts.

2.1L2L3L4step 1.1∎

Since the source and target are additive by [L2], those finite products and coproducts are finite biproducts. Therefore step 1.1 lets [L3] conclude that F is additive. Together with step 1.1, that is exactly the definition of exactness in [L4].

Depends on

Used by

Dependency tree · two levels

19 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