ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30

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.

The connecting morphism computed for a short exact sequence of abelian groups

Example

In $A b$ , consider

The connecting morphism $δ : \ker (0) ≅ Z / 2 ⟶ coker (\times 2) ≅ Z / 2$ is the identity map.

Facts & Assumptions

Given: The diagram above in $A b$ .

[L1]

Abelian groups form an abelian category (Abelian groups form an abelian category).

[L2]

The connecting morphism exists, and the snake sequence is exact (The connecting morphism exists and is unique, Snake lemma in an abelian category).

Verification

1.1L1algebra

By [L1], the displayed diagram is valid snake data. Its kernel and cokernel terms are $\ker (\times 2) = 0, \ker (0) = Z / 2, coker (\times 2) = Z / 2.$

2.1L2step 1.1algebra

The snake sequence from [L2] therefore reduces to $0 \to 0 \to 0 \to Z / 2 \overset{δ}{\to} Z / 2 \to Z / 2 \to Z / 2 \to 0.$ Exactness at the source and target of $δ$ forces $δ$ to be both injective and surjective, hence the identity automorphism of $Z / 2$ .

3.1step 2.1∎

So in this concrete diagram the connecting morphism is the identity on $Z / 2$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

Saunders Mac Lane, Categories for the Working Mathematician, Lemma VIII.4.5 (standard reference, not scraped)