CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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.

The injection $\emptyset \to {*}$ is monic in $S e t$ but is not split

Statement refuted

The assertion that every monomorphism is a split monomorphism is false.

Facts & Assumptions

Given: The unique function $i : \emptyset \to {*}$ .

[L1]

Monomorphisms in $S e t$ are exactly injections (In $S e t$ , monomorphisms are exactly injections and epimorphisms are exactly surjections).

[L2]

A split monomorphism $i : A \to B$ requires a retraction $r : B \to A$ with $r i = 1_{A}$ (Split monomorphism, split epimorphism, retraction, and section).

Counterexample

technique · direct

1.1

The map $i$ is injective because its domain has no elements, so [L1] makes it a monomorphism.

1.2

There is no function $r : {*} \to \emptyset$ , since the value $r (*)$ would have to be an element of the empty set.

given

2.1

Therefore $i$ admits no retraction and is not split by [L2], although it is monic by step 1.1.

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

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.

The injection ∅→{∗}∅→{∗} is monic in SetSet but is not split