DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Cartesian closed category

Definition

A category $C$ is cartesian closed when:

$C$ has finite products; and
for each object $B$ , the functor $- \times B$ has a right adjoint.

Writing that right adjoint as $(-)^{B}$ , the object $C^{B}$ is the exponential object of $C$ by $B$ in the sense of Exponential object. Since a category with finite products is canonically monoidal under the cartesian product (A category with finite products is monoidal), a cartesian closed category is exactly a cartesian monoidal category whose tensor product is closed.

Depends on

Used by

In a cartesian closed category, any initial object is strict Corollary
Locally cartesian closed category Definition
FALSE: every cartesian closed category has all finite limits False statement
FALSE: every cartesian closed category is locally cartesian closed False statement
A cartesian closed preorder has relative implications Theorem
A presheaf category on a small category is cartesian closed Theorem
Currying and uncurrying are mutually inverse Theorem
Set is cartesian closed Theorem
The category of small categories is cartesian closed Theorem

Dependency tree · two levels

7 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., Definition 4.4.10 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, 2nd ed., IV.6 (standard reference, not scraped)