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

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FALSE: every cartesian closed category has all finite limits

Statement

Every cartesian closed category has all finite limits.

Facts & Assumptions

Given: The full subcategory ${S e t}_{\neq \emptyset}$ of $S e t$ on the nonempty sets.

[L1]

A cartesian closed category is required to have finite products and exponentials, but not arbitrary finite limits as part of the definition (Cartesian closed category).

[A1]

If $A, B$ are nonempty sets, then $A \times B$ and $B^{A}$ are again nonempty, so ${S e t}_{\neq \emptyset}$ is cartesian closed.

Refutation

technique · direct

1.1

By [A1], the category ${S e t}_{\neq \emptyset}$ has products and exponentials for all of its objects, so by [L1] it is cartesian closed.

givenL1A1

2.1

Consider the two constant maps $1 ⇉ 2$ with values $0$ and $1$ . Their equalizer in $S e t$ is the empty set, which is not an object of ${S e t}_{\neq \emptyset}$ . So that equalizer does not exist in ${S e t}_{\neq \emptyset}$ .

step 1.1givenalgebra

3.1

Therefore a cartesian closed category need not have all finite limits. The statement is false.

step 2.1∎

Depends on

Cartesian closed category

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

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