TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)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 is self-dual

Statement

A composable pair $A \overset{f}{\to} B \overset{g}{\to} C$ is exact at $B$ in an abelian category $A$ if and only if the opposite pair $C \overset{g^{o p}}{\to} B \overset{f^{o p}}{\to} A$ is exact at $B$ in $A^{o p}$ .

Facts & Assumptions

Given: A composable pair $A \overset{f}{\to} B \overset{g}{\to} C$ in an abelian category $A$ .

[L1]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

[L2]

Exactness at $B$ means either $[im (f)] = [\ker (g)]$ or, equivalently, $[coker (f)] = [coim (g)]$ (Exactness at a node, Image and coimage in a category with kernels and cokernels).

Proof

technique · direct

1.1

By [L1], the opposite category $A^{o p}$ is again abelian, so the definition [L2] applies there. Passing to the opposite exchanges kernels with cokernels and images with coimages.

L1L2

2.1

Therefore the equality $[im (f)] = [\ker (g)]$ in $A$ is exactly the equality $[im (g^{o p})] = [\ker (f^{o p})]$ in $A^{o p}$ . By [L2], those are the two exactness assertions.

L2step 1.1∎

Depends on

Used by

Dependency tree · two levels

7 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 (standard reference, not scraped)
David Mehrle, Category Theory, Part III, Chapter 7 (standard reference, not scraped)