TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01‡ 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.

The category of small categories is cartesian closed

Statement

The category $C a t$ of small categories is cartesian closed. For small categories $D, E$ , the exponential object of $E$ by $D$ is the functor category $[D, E]$ .

Facts & Assumptions

Given: Small categories $C, D, E$ .

[L1]

$C a t$ is cartesian monoidal, so it has finite products (Set, Cat, and every complete category are cartesian monoidal).

[L2]

For small source and locally small target, the functor category $[D, E]$ exists and is locally small; if both are small, then it is small (Functor category $[C, D]$ , If $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small).

[L3]

A cartesian closed category is one with finite products and a right adjoint to each functor $- \times D$ (Cartesian closed category).

Proof

technique · direct

1.1

By [L1], $C a t$ has the finite products required in [L3]. By [L2], the functor category $[D, E]$ is again a small category.

givenL1L2

1.2

A functor $F : C \times D \to E$ determines a functor $\tilde{F} : C \to [D, E]$ by $\tilde{F} (c) (d) = F (c, d)$ on objects and similarly on morphisms. Conversely, a functor $Φ : C \to [D, E]$ determines $\hat{Φ} : C \times D \to E$ by evaluation, $\hat{Φ} (c, d) = Φ (c) (d)$ . These assignments are inverse and natural in $C$ .

givenL2algebra

2.1

Step 1.2 is exactly the adjunction $C a t (C \times D, E) ≅ C a t (C, [D, E])$ . Thus $- \times D$ has right adjoint $[D, -]$ . With step 1.1, [L3] now gives that $C a t$ is cartesian closed.

step 1.1step 1.2L3∎

Depends on

Used by

The exponential of two small categories computed on a walking-arrow source Example

Dependency tree · two levels

14 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

Tom Leinster, Basic Category Theory, Example 6.3.17 (standard reference, not scraped)