False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

FALSE: every cartesian closed category is locally cartesian closed

Statement

Every cartesian closed category is locally cartesian closed.

Facts & Assumptions

Given: The category ${S e t}_{\neq \emptyset}$ of nonempty sets.

[L1]

Local cartesian closedness requires every slice to be cartesian closed (Locally cartesian closed category).

[A1]

The category ${S e t}_{\neq \emptyset}$ is cartesian closed, by the same function-set argument as in the previous false statement.

Refutation

technique · direct

1.1givenA1

By [A1], ${S e t}_{\neq \emptyset}$ is cartesian closed.

2.1step 1.1givenalgebra

In the slice over the two-point set $2 = {0, 1}$ , consider the singleton inclusions ${0} ↪ 2$ and ${1} ↪ 2$ . Their product in the slice would be their pullback, which is the empty set over $2$ . That object is not present in ${S e t}_{\neq \emptyset} / 2$ .

3.1step 2.1L1∎

So the slice over $2$ is not even finitely complete, hence not cartesian closed. Therefore ${S e t}_{\neq \emptyset}$ is not locally cartesian closed, and the statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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, 2nd ed., Section 4.6 (standard reference, not scraped)