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

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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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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

Used by

FALSE: member equivalence is transitive in any pointed category with pullbacks False statement
Each object has a zero member and each member has a negative Proposition
The cost of the member calculus Remark
Four lemma in an abelian category Theorem
Members modulo equivalence correspond to subobjects Theorem
Nine lemma in an abelian category Theorem
Nine lemma variants by which rows are assumed exact Theorem
Sharp nine lemma Theorem
Short five lemma in an abelian category Theorem
The subtraction surrogate Theorem
Weak four lemma with the exactness hypotheses named Theorem

Dependency tree · two levels

11 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, Proposition 2 and Theorem 3 (standard reference, not scraped)