nLab cartesian closed enriched category

Idea

A cartesian closed enriched category is an enriched category that is simultaneously cartesian monoidal and monoidal closed.

Definitions

The generalization of closed monoidal category to the enriched setting is standard.

There are multiple inequivalent definitions we could make for what it means for an enriched category to be cartesian monoidal.

Definition

In Street 14, a cartesian monoidal enriched category is an enriched monoidal category $(C,\otimes, I)$ such that the functors $I : I \to C$ and $(-\otimes -) : C\otimes C \to C$ have $V$-enriched left adjoints.

A cartesian closed enriched category is a cartesian monoidal enriched category in which all the $V$-functors $(X\otimes -) \colon C \to C$ and $(-\otimes X) \colon C \to C$ have $V$-enriched right adjoints.

Definition

If we want the monoidal product to literally be the categorical product, we need, in general, $V$ to be semicartesian.

Let $V$ be a semicartesian 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 that $C$ 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

For the following, we use the second definition given above.

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

• Ross Street, Kan extensions and cartesian monoidal categories arXiv:1409.6405