False statementConstruction: AI-adaptedVerification: AI-generatedSession-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: a functor preserving binary products and equalizers must preserve all finite limits

Statement refuted

Every functor that preserves binary products and equalizers preserves all finite limits.

Facts & Assumptions

Given: The terminal category $1$ and the functor $F : 1 \to S e t$ sending its sole object to $\emptyset$ .

[L1]

Finite-limit criteria require nullary product data, equivalently a terminal object, in addition to binary products and equalizers (Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals).

[F1]

Continuous means preserving all small limits (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

[F2]

Morphisms of $S e t$ are functions (Sets and functions form the large locally small category $S e t$ ).

Refutation

technique · constant-empty functor

1.1

The binary product of the sole object of $1$ with itself is that object, and $\emptyset \times \emptyset = \emptyset$ . The image product cone consists of identity functions on $\emptyset$ , so $F$ preserves the binary product.

1.2

Every parallel pair in $1$ is $(1, 1)$ and has identity equalizer. Its image is $(1_{\emptyset}, 1_{\emptyset})$ , whose identity is also an equalizer. Thus $F$ preserves equalizers.

1.3

The sole object of $1$ is terminal, but $\emptyset$ is not terminal in $S e t$ because no function ${*} \to \emptyset$ exists. So $F$ does not preserve the empty product, hence does not preserve all finite limits and is not continuous by [F1].

F1F2

2.1

This refutes the statement and exhibits exactly the missing nullary case in [L1].

L1step 1.1step 1.2step 1.3∎

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: 38 results over 15 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, Theorem 3.5.17 (standard reference, not scraped)