TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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The quotient by a subobject is independent of the chosen representing monomorphism

Statement

If two monomorphisms $m : M \to A$ and $n : N \to A$ represent the same subobject of $A$ , then their cokernels are canonically isomorphic. Hence the notation $A / B$ depends only on the subobject class $[B \to A]$ .

Facts & Assumptions

Given: Two monomorphisms $m : M \to A$ and $n : N \to A$ representing the same subobject.

[L1]

The quotient by a subobject is defined as the cokernel of a representing monomorphism (The quotient of an object by a subobject).

[L2]

Representing the same subobject means mutual factorization (Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it).

[L3]

The cokernel assignment depends only on the subobject class and gives the inverse order-anti-isomorphism to the kernel assignment (Kernel and cokernel are mutually inverse order-preserving correspondences between subobjects and quotient objects).

Proof

technique · direct

1.1

By [L2], the condition that $m$ and $n$ represent the same subobject is exactly $[m] = [n]$ . Applying [L3] gives $[coker (m)] = [coker (n)]$ as quotient-object classes.

L2L3

2.1

Equality of quotient-object classes means the two cokernels are joined by a unique compatible isomorphism. By [L1], that is exactly the claim that the quotient $A / B$ is independent of the chosen representing monomorphism.

L1step 1.1∎

Depends on

Used by

First isomorphism theorem in an abelian category Theorem
Third isomorphism theorem in an abelian category Theorem

Dependency tree · two levels

8 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

Junhan Tan, The Freyd-Mitchell Embedding Theorem, Theorem 2.3 (standard reference, not scraped)