TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-29

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Members modulo equivalence correspond to subobjects

Statement

Let $A$ be an object of an abelian category. Sending a member $x : X \to A$ to the subobject of $A$ represented by its image inclusion induces a bijection between equivalence classes of members of $A$ and subobjects of $A$ .

Facts & Assumptions

Given: A member $x : X \to A$ and, when needed, a second member $y : Y \to A$ .

[L1]

Every morphism factors as an epimorphism followed by a monomorphism (Every morphism factors as an epimorphism followed by a monomorphism, uniquely up to unique isomorphism).

[L2]

Subobjects are mutual-factorization classes of monomorphisms (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms, Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it).

[L4]

Member equivalence is transitive (Equivalence of members, Member equivalence is transitive).

Proof

technique · direct

1.1L1L2construct

Factor $x$ as $X \overset{e_{x}}{\to} I_{x} \overset{m_{x}}{\to} A$ with $e_{x}$ epic and $m_{x}$ monic, using [L1], and assign to the class of $x$ the subobject $[⁣ m_{x} ⁣]$ of $A$ .

1.2L1L2

Every subobject is represented. If $m : S ↣ A$ is monic, then $m$ itself is a member of $A$ , and the factorization $S \overset{1_{S}}{\to} S \overset{m}{\to} A$ shows that its image class is $[⁣ m ⁣]$ .

2.1L1L2L4step 1.1

This assignment is well defined on equivalence classes. The equality $x 1_{X} = m_{x} e_{x}$ with epic maps on the right shows $x \equiv m_{x}$ , and similarly $y \equiv m_{y}$ . If $x \equiv y$ , then transitivity from [L4] gives $m_{x} \equiv m_{y}$ , which for monomorphisms into $A$ is exactly equality of subobject classes by [L2].

3.1L2L4step 2.1

The assignment is injective. If $x$ and $y$ determine the same subobject, then $[⁣ m_{x} ⁣] = [⁣ m_{y} ⁣]$ by [L2]. Step 2.1 gives $x \equiv m_{x}$ and $y \equiv m_{y}$ , while equality of subobjects makes $m_{x}$ and $m_{y}$ equivalent as members. Another use of [L4] yields $x \equiv y$ .

4.1step 1.2step 3.1∎

Therefore the assignment of step 1.1 is a bijection from member-equivalence classes to subobjects of $A$ .

Depends on

Used by

A general member of an abelian group need not come from an element Counterexample
An ordinary element as the member from the integers Example
Members of an abelian group correspond to its subgroups Example
Epimorphy is detected by members Theorem
Exactness is detected by members Theorem

Dependency tree · two levels

18 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, VIII.4 (standard reference, not scraped)