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.

A presheaf category on a small category is cartesian closed

Statement

Let $C$ be a small category and let $\hat{C} = {S e t}^{C^{o p}}$ . Then $\hat{C}$ is cartesian closed. For presheaves $P, Q$ , one exponential object is the presheaf defined by

$(Q^{P}) (c) : = Nat (y (c) \times P, Q),$

with restriction along $f : d \to c$ given by precomposition with $y (f) \times 1_{P}$ .

Facts & Assumptions

Given: A small category $C$ and presheaves $H, P, Q$ on $C$ .

[L1]

The Yoneda embedding sends $c$ to the representable presheaf $y (c)$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding).

[L2]

For any presheaf $R$ , natural transformations $y (c) \to R$ are naturally in bijection with elements of $R (c)$ (For a presheaf $P$ , $Nat (C (-, a), P) ≅ P (a)$ naturally in $a$ and $P$ ).

[L3]

In $S e t$ , currying gives $S e t (A \times B, C) ≅ S e t (A, C^{B})$ (Currying gives the adjunction $- \times A ⊣ (-)^{A}$ in $S e t$ ).

[L4]

Products of presheaves are computed pointwise, and the presheaf category is locally small because $C$ is small (For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise, If $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small).

[L5]

A cartesian closed category has finite products and exponentials (Cartesian closed category).

Proof

technique · direct

1.1

By [L4], binary products in $\hat{C}$ are computed pointwise, so $H \times P$ is the presheaf with $(H \times P) (c) = H (c) \times P (c)$ . The assignment $c \mapsto Nat (y (c) \times P, Q)$ with restriction by precomposition along $y (f) \times 1_{P}$ is therefore a presheaf $Q^{P}$ .

givenL1L4algebra

2.1

Given $α : H \to Q^{P}$ , define $\tilde{α} : H \times P \to Q$ by sending $(h, p) \in H (c) \times P (c)$ to the element of $Q (c)$ corresponding under [L2] to the natural transformation $y (c) \overset{h}{\to} H \overset{α}{\to} Q^{P}$ , evaluated at $p$ . Conversely, given $β : H \times P \to Q$ , use [L2] and the set-level currying of [L3] objectwise to define ${\hat{β}}_{c} (h) \in Nat (y (c) \times P, Q)$ ; naturality in $c$ is exactly the restriction rule from step 1.1.

givenL2L3L4construct

3.1

The two constructions of step 2.1 are inverse because Yoneda identifies a natural transformation out of $y (c)$ with its value at $1_{c}$ , and the set-level currying and uncurrying in [L3] are inverse. Hence $\hat{C} (H \times P, Q) ≅ \hat{C} (H, Q^{P})$ naturally in $H$ .

step 2.1L2L3algebra

4.1

Step 3.1 shows that $- \times P$ has right adjoint $(-)^{P}$ , while [L4] gives the finite products. Therefore [L5] implies that $\hat{C}$ is cartesian closed.

step 1.1step 3.1L4L5∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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 4.4.11 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Exercise 6.3.25 (standard reference, not scraped)