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.

Left-closed, right-closed, and biclosed monoidal categories

Definition

Let $(C, \otimes, 1)$ be a monoidal category (Monoidal category).

It is right closed when for every object $X$ the functor $- \otimes X : C \to C$ has a right adjoint in the sense of Adjunction by unit, counit, and the triangle identities. A chosen right adjoint is written $[X, -]$ .
It is left closed when for every object $X$ the functor $X \otimes - : C \to C$ has a right adjoint. A chosen right adjoint is written $⟨ X, - ⟩$ .
It is biclosed when it is both left closed and right closed.

Thus a right-closed structure gives natural bijections

$C (A \otimes X, Y) ≅ C (A, [X, Y]),$

and a left-closed structure gives natural bijections

$C (X \otimes A, Y) ≅ C (A, ⟨ X, Y ⟩) .$

This page keeps the two closures separate unless a symmetry is supplied later.

Depends on

Used by

A monoidal category need not be closed Counterexample
The internal hom and its evaluation morphism Definition
FALSE: the left and right internal homs agree in every monoidal category False statement
In a biclosed monoidal category tensor is cocontinuous in each variable Theorem
The internal hom is unique up to a unique adjunction-compatible natural isomorphism 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 4.4.7 (standard reference, not scraped)
G. M. Kelly, Basic Concepts of Enriched Category Theory, Section 1.5 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, 2nd ed., VII.7 (standard reference, not scraped)