TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-08-29

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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.

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Member equivalence is transitive

Statement

For members of one object in an abelian category, the relation $\equiv$ is transitive.

Facts & Assumptions

Given: An abelian category and members $x : X \to A$ , $y : Y \to A$ , and $z : Z \to A$ with $x \equiv y$ and $y \equiv z$ .

[L1]

The relation $x \equiv y$ is witnessed by epimorphisms from one common domain as in Equivalence of members.

[L2]

Pullbacks exist in an abelian category (Pullbacks and pushouts as limits and colimits of cospans and spans).

[L3]

The pullback of an epimorphism is an epimorphism (The pullback of an epimorphism is an epimorphism).

Proof

technique · direct

1.1L1choose

Choose epimorphisms $u : W_{1} \to X$ , $v : W_{1} \to Y$ , $w : W_{2} \to Y$ , and $r : W_{2} \to Z$ with $x u = y v$ and $y w = z r$ , using [L1].

1.2L2L3construct

Form the pullback of $v$ and $w$ , with projections $v^{'} : P \to W_{1}$ and $w^{'} : P \to W_{2}$ . By [L3], both $v^{'}$ and $w^{'}$ are epic.

2.1L1step 1.1step 1.2algebra

The pullback equation gives $v v^{'} = w w^{'}$ , so $x u v^{'} = y v v^{'} = y w w^{'} = z r w^{'}$ . Hence the epimorphisms $u v^{'} : P \to X$ and $r w^{'} : P \to Z$ witness $x \equiv z$ .

3.1step 2.1∎

Therefore member equivalence is transitive.

Depends on

Dependency tree · 2 levels

Sources

Saunders Mac Lane, Categories for the Working Mathematician, Proposition 2 and Theorem 3