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A V-category is tensored exactly when each covariant hom has a left enriched adjoint

Statement

Assume $V$ is right closed monoidal. A $V$ -category $B$ is tensored if and only if, for every object $C$ of $B$ , the covariant enriched hom-functor $B (C, -) : B \to V$ has a left enriched adjoint.

Facts & Assumptions

Given: A right-closed monoidal base $V$ , a $V$ -category $B$ , and an object $C$ of it.

[L1]

Tensors of $C$ by objects $X \in V$ are characterized by $B (X \otimes C, B) ≅ [X, B (C, B)]$ (Tensor and cotensor in a V-category).

[L2]

An enriched adjunction is exactly a natural isomorphism of enriched hom-objects (Enriched adjunction).

[L3]

The functor $B (C, -)$ is the representable enriched functor at $C$ (Representable enriched functor).

Proof

technique · direct

1.1

If $B$ is tensored, then for each $X \in V$ and $B \in B$ the tensor formula of [L1] is exactly the enriched adjunction isomorphism between the functor $X \mapsto X \otimes C$ and the representable functor $B (C, -)$ from [L3]. So $B (C, -)$ has a left enriched adjoint.

L1L2L3given

1.2

Conversely, suppose $B (C, -)$ has a left enriched adjoint $L_{C}$ . Then [L2] gives isomorphisms $B (L_{C} X, B) ≅ [X, B (C, B)]$ natural in $X$ and $B$ . Comparing with [L1], the object $L_{C} X$ is exactly the tensor $X \otimes C$ . So tensors exist for every $X$ and $C$ .

L1L2L3algebra

2.1

Hence $B$ is tensored exactly when each covariant hom-functor has a left enriched adjoint.

step 1.1step 1.2∎

Depends on

Used by

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Dependency tree · two levels

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Sources

G. M. Kelly, Basic Concepts of Enriched Category Theory, Section 3.7 (standard reference, not scraped)