CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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.

Additive categories are closed under passage to the opposite

Statement

If $C$ is additive, then $C^{o p}$ is additive.

Facts & Assumptions

Given: An additive category $C$ .

[L1]

The opposite of a preadditive category is preadditive (The opposite of a preadditive category is preadditive).

[L2]

An additive category is a preadditive category with finite biproducts (Additive category).

Proof

technique · direct

1.1L1L2

By [L2], $C$ is preadditive, so $C^{o p}$ is preadditive by [L1].

2.1L2step 1.1

By [L2], $C$ has finite biproducts. Passing to the opposite category swaps products and coproducts, so every finite biproduct diagram in $C$ becomes a diagram in $C^{o p}$ that is again simultaneously a finite product and a finite coproduct. Thus $C^{o p}$ has finite biproducts.

3.1L2step 2.1∎

Thus $C^{o p}$ is preadditive and has finite biproducts, so it is additive by [L2].

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Section 12.3: Preadditive and additive categories