CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

An additive category is an Ab-enriched category with a zero object and finite biproducts

Statement

Every additive category is an $A b$ -enriched category whose underlying ordinary category has a zero object and all finite biproducts.

Facts & Assumptions

Given: An additive category $A$ .

[L1]

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

[L2]

A category is $A b$ -enriched exactly when it is preadditive (Ab-enriched categories are exactly preadditive categories).

Proof

technique · direct

1.1L1given

By [L1], $A$ is preadditive and has finite biproducts; in particular it has a zero object as part of that finite-biproduct structure.

2.1L2step 1.1

The preadditivity from step 1.1 is exactly the hypothesis of [L2], so $A$ is $A b$ -enriched. The zero object and finite biproducts remain part of the underlying ordinary category because [L2] only rephrases the hom-structure.

3.1step 2.1∎

Therefore every additive category has the stated enriched form.

Depends on

Used by

Projective epimorphisms onto the simples generate every finite-length object Lemma

Dependency tree · two levels

9 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

Emily Riehl, Categorical Homotopy Theory, Section 3.3 (standard reference, not scraped)
The Stacks Project, Section 12.3 (standard reference, not scraped)