nLab cartesian closed enriched category

Definition

Let $V$ be a monoidal category with products and $C$ a $V$-enriched category with products, in the enriched sense that we have a $V$-natural isomorphisms of hom-objects in $V$:

$$C(Z, X\times Y) \;\cong\; C(Z,X) \times C(Z,Y) \,.$$

We say $C$ is $V$-cartesian-closed if each $V$-functor $(X\times -) \colon C \to C$ has a $V$-enriched right adjoint.

Relation to ordinary cartesian closedness

If $C$ is $V$-cartesian-closed, then its underlying ordinary category $C_0$ is cartesian closed in the usual sense, since $V$-enriched right adjoints have underlying ordinary right adjoints.

The converse is true in some cases, such as the following:

• When $V=Set$, trivially.

• More generally, whenever the underlying-set functor $V(I,-) : V\to Set$ is conservative, since the morphism of hom-objects $C(X,Y^Z) \to C(X\times Y,Z)$ induced by the evaluation morphism $Y^Z\times Y \to Z$ has invertible image in $Set$, hence is itself invertible if $V(I,-)$ is conservative.

• When $V$ is cartesian monoidal and $C=V$: a cartesian closed category is automatically enriched-cartesian-closed over itself. In other words, the defining isomorphisms $V_0(X\times Y,Z) \cong V_0(X, Z^Y)$ induce, by the Yoneda lemma, isomorphisms of exponential objects $Z^{X\times Y} \cong (Z^Y)^X$.

However, the converse is false in general. Counterexamples can be found in this mathoverflow discussion.

Related concepts

• cartesian closed category

• locally cartesian closed enriched category

• monoidal enriched category

• cartesian closed model category

References

• Enriched cartesian closed categories on MO

• Simplicial cartesian closed categories on MO