TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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A closed monoidal category is enriched in itself

Statement

If $V$ is a right-closed monoidal category whose object collection is a set, then it carries a canonical $V$ -enriched category structure with the same objects as $V$ and hom-object $\underline{V} (X, Y) : = [X, Y] .$ The enriched composition is the internal-hom composition morphism and the enriched identity is the unit morphism into $[X, X]$ .

Facts & Assumptions

Given: A right-closed monoidal category $V$ with a set of objects.

[L1]

Right closedness supplies internal hom-objects $[X, Y]$ (Left-closed, right-closed, and biclosed monoidal categories).

[L2]

There is a natural composition morphism $[Y, Z] \otimes [X, Y] \to [X, Z]$ with associative and unital laws (The internal-hom composition morphism).

[L3]

The unit object yields the external hom-set bijection $V (1, [X, Y]) ≅ V (X, Y)$ (The tensor unit is an internal-hom unit).

Proof

technique · direct

1.1

By [L1], every ordered pair $(X, Y)$ has an internal hom-object $[X, Y]$ . Take that object to be the enriched hom-object $\underline{V} (X, Y)$ .

L1given

1.2

Use the composition morphism of [L2] as the enriched composition map $\underline{V} (Y, Z) \otimes \underline{V} (X, Y) \to \underline{V} (X, Z)$ . For each $X$ , use the unit morphism $1 \to [X, X]$ from [L2] as the enriched identity. The associativity and unit axioms required by Enriched category over a monoidal base are exactly the associativity and unit laws already proved in [L2].

L2algebra

2.1

Thus the data of steps 1.1 and 1.2 satisfy the definition of a $V$ -category. The bijection of [L3] explains why the global elements of the enriched hom-object recover the ordinary morphisms of $V$ .

L3step 1.1step 1.2∎

Depends on

Used by

Enriched adjunction Definition
Enriched weighted limit Definition
Representable enriched functor Definition
Tensor and cotensor in a V-category Definition
Strong enriched Yoneda lemma as a particular end Theorem

Dependency tree · two levels

8 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

G. M. Kelly, Basic Concepts of Enriched Category Theory, equations (1.28) to (1.32) (standard reference, not scraped)
Emily Riehl, Categorical Homotopy Theory, Section 3.2 (standard reference, not scraped)