DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-29

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

Definition

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

Used by

Exact sequence and short exact sequence in an abelian category Definition
FALSE: the subobject-side definition of exactness needs no canonical image monomorphism False statement
A short exact sequence is a kernel-cokernel pair Theorem
Degenerate exactness criteria Theorem
Exactness is detected by members Theorem
Exactness is self-dual Theorem
The arrow-theoretic criterion for exactness Theorem
The covering criterion for exactness Theorem
The subobject inequalities underlying exactness Theorem

Dependency tree · two levels

6 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

The Stacks Project, Section 12.5, Definition 12.5.7 (standard reference, not scraped)
David Mehrle, Category Theory, Part III, Definition 7.20 (standard reference, not scraped)