CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Two different monomorphisms can represent the same subobject

Counterexample

Into $X = {0, 1}$ , the maps $m : {0} \to X$ , $m (0) = 0$ , and $n : {*} \to X$ , $n (*) = 0$ , are distinct monomorphisms representing the same subobject.

Facts & Assumptions

Given: The displayed maps in $S e t$ .

[L1]

Subobjects are equivalence classes of monomorphisms under mutual factorisation (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms).

[L2]

The factor maps between mutually factoring monomorphisms are unique inverse isomorphisms (Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it).

Verification

technique · direct

1.1

Both $m$ and $n$ are injective, and an injection $f$ in $S e t$ is monic: $f \circ g = f \circ h$ gives $f (g (x)) = f (h (x))$ for every $x$ , hence $g (x) = h (x)$ and $g = h$ . They are not the same map, since their domains ${0}$ and ${*}$ are distinct.

given

2.1

Let $u : {0} \to {*}$ and $v : {*} \to {0}$ be the unique maps. Then $m = n \circ u$ and $n = m \circ v$ , so the monomorphisms mutually factor; [L2] also identifies $u$ and $v$ as inverse isomorphisms.

step 1.1L2

3.1

By [L1], $[m] = [n]$ even though $m \neq n$ .

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.