False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: every category has all small limits

Statement refuted

Every category has all small limits.

Facts & Assumptions

Given: The category of nonempty sets and all functions.

[F1]

A category is complete when every small diagram in it has a limit (Finite, small, and large limits and colimits; complete and cocomplete categories).

[F2]

An equalizer of $f, g : X ⇉ Y$ must receive every map on which $f$ and $g$ agree (Equalizers and coequalizers as limits and colimits of a parallel pair).

Refutation

technique · counterexample

1.1

Let $f, g : {*} ⇉ {0, 1}$ be constant at $0$ and $1$ . For any nonempty set $X$ , the unique function $X \to {*}$ has composites constant at different values, so it does not equalize $f, g$ .

given

2.1

Thus this parallel-pair diagram has no cone and in particular no equalizer [F2]. Its indexing category is finite and hence small.

F2step 1.1

3.1

By [F1], the category of nonempty sets is not complete. This category refutes the universal statement.

F1step 2.1∎

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: 13 results over 5 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.

Sources

E. Riehl, Category Theory in Context, Definition 3.2.1 (standard reference, not scraped)