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

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Set is cartesian closed

Statement

The category $S e t$ is cartesian closed. For sets $B, C$ , an exponential object of $C$ by $B$ is the function set $C^{B} = {Hom}_{S e t} (B, C)$ with evaluation $(f, b) \mapsto f (b)$ .

Facts & Assumptions

Given: Sets $A, B, C$ .

[L1]

A cartesian closed category has finite products and exponentials, equivalently a right adjoint to each product functor $- \times B$ (Cartesian closed category).

[L2]

$S e t$ has all small limits, hence in particular finite products (Set has all small limits, realized as compatible tuples in a set-indexed product).

[L3]

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

Proof

technique · direct

1.1givenL2

By [L2], $S e t$ has finite products.

1.2givenL3

Let $C^{B}$ be the set of functions $B \to C$ , and let $ev : C^{B} \times B \to C$ send $(f, b)$ to $f (b)$ . By [L3], every map $A \times B \to C$ corresponds naturally and bijectively to a map $A \to C^{B}$ . This is exactly the universal property of an exponential object.

2.1step 1.1step 1.2L1∎

Step 1.1 gives the finite-product part of [L1], and step 1.2 gives the right adjoint to $- \times B$ for each $B$ . Therefore $S e t$ is cartesian closed.

Depends on

Used by

A function set computed as an exponential object Example

Dependency tree · two levels

13 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., Example 4.4.9 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, 2nd ed., IV.6 (standard reference, not scraped)