False statementConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

FALSE: the subobject-side definition of exactness needs no canonical image monomorphism

Statement

One may define exactness of $A \overset{f}{\to} B \overset{g}{\to} C$ by treating $im (f)$ merely as an object, without specifying its canonical monomorphism into $B$ , and writing the purported subobject equality $[im (f)] = [\ker (g)]$ anyway.

Facts & Assumptions

Given: The claim of the statement.

[L1]

Exactness at a node is stated as equality of the image subobject of $f$ with the kernel subobject of $g$ (Exactness at a node).

[L2]

A subobject of $B$ is represented by a monomorphism into $B$ (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms), and the image of $f$ includes its defining kernel arrow into $B$ (Image and coimage in a category with kernels and cokernels).

Refutation

technique · direct

1.1L1L2

By [L2], an object alone does not represent a subobject of $B$ : the structure monomorphism into $B$ is essential data. Thus the proposed equality is not a typed equality of subobjects until the canonical image monomorphism has been specified.

2.1step 1.1∎

Therefore the statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

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

Peter Freyd, Abelian Categories, Theorem 2.21 (standard reference, not scraped)
Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)