DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-29

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.

Exactness at a node

Let $A \overset{f}{\to} B \overset{g}{\to} C$ be composable morphisms in an abelian category.

The pair is exact at $B$ when the image of $f$ and the kernel of $g$ represent the same subobject of $B$ : $[im (f)] = [\ker (g)] .$

Equivalently, by the dual description in the opposite abelian category (The opposite of an abelian category is abelian) together with the definitions of image and coimage (Image and coimage in a category with kernels and cokernels), the pair is exact at $B$ exactly when the cokernel of $f$ and the coimage of $g$ represent the same quotient of $B$ : $[coker (f)] = [coim (g)] .$

The well-definedness of the first equality as a comparison of subobjects is exactly the content of The subobject inequalities underlying exactness ↗.

Depends on