TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

A locally cartesian closed category with a terminal object is cartesian closed

Statement

If $C$ is locally cartesian closed and has a terminal object $1$ , then $C$ is cartesian closed.

Facts & Assumptions

Given: A locally cartesian closed category $C$ with terminal object $1$ .

[L1]

Every slice $C / X$ of a locally cartesian closed category is cartesian closed (Locally cartesian closed category).

[L2]

A terminal object receives a unique morphism from every object (Initial object, terminal object, and zero object).

Proof

technique · direct

1.1

The assignment $A \mapsto!_{A} : A \to 1$ defines a functor $C \to C / 1$ , and forgetting the structure map defines a functor back. Because [L2] gives a unique map to $1$ , these two functors are inverse isomorphisms of categories.

givenL2

2.1

By [L1], the slice $C / 1$ is cartesian closed. Transport that structure across the isomorphism of step 1.1 to conclude that $C$ itself is cartesian closed.

step 1.1L1

3.1

Therefore a locally cartesian closed category with a terminal object is cartesian closed.

step 2.1∎

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.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Lemma 4.6.3(ii) (standard reference, not scraped)