CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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.

Pointed sets are not additive

Statement refuted

Refuted claim: the category of pointed sets is additive.

Take $A = B = {0, 1}$ with basepoint $0$ .

Facts & Assumptions

Given: The pointed sets $A = B = {0, 1}$ with basepoint $0$ .

[A1]

In ${S e t}_{*}$ , the wedge $A \lor B$ has three points and the product $A \times B$ has four points.

Counterexample

technique · direct

1.1

The canonical map $A \lor B \to A \times B$ sends the three wedge points to $(0, 0)$ , $(1, 0)$ , and $(0, 1)$ , so it misses $(1, 1)$ . Therefore the wedge and product are not a biproduct pair.

2.1

An additive category must in particular have binary biproducts, so ${S e t}_{*}$ cannot be additive.

step 1.1∎

Depends on

FALSE: finite products and finite coproducts already force biproducts

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Emily Riehl, Category Theory in Context, Chapter 3 (standard reference, not scraped)