DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-28

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 quotient of an object by a subobject

Definition

Let $[m]$ be a subobject of an object $A$ in an abelian category, represented by a monomorphism $m : M \to A$ . The quotient of $A$ by $[m]$ is the quotient object represented by the cokernel of $m$ :

$A / [m] : = coker (m) .$

Because every monomorphism is the kernel of its cokernel (Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel), this quotient is the object paired with the given subobject by the kernel-cokernel correspondence. The next item checks that the definition does not depend on the chosen representative of the class $[m]$ .

Depends on

Dependency tree · 2 levels

Sources

Junhan Tan, The Freyd-Mitchell Embedding Theorem, Theorem 2.3