nLab locally cartesian closed enriched category

Definition

Let $V$ be a monoidal category and $C$ a $V$-enriched category with pullbacks in the enriched sense. Then for any morphism $f:x\to y$ in (the underlying ordinary category of) $C$, there is a pullback $V$-functor $f^*:C/y \to C/x$ between the enriched slice categories, and each enriched slice category $C/x$ has $V$-enriched products. We say $C$ is $V$-locally-cartesian-closed if the following equivalent conditions hold:

• Each $V$-functor $f^*$ has a right $V$-adjoint $f_*$.
• Each $V$-category $C/x$ is $V$-cartesian closed.

Relation to ordinary local cartesian closure

If $C$ is $V$-locally-cartesian-closed, then its underlying ordinary category $C_0$ is locally 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.

• When $V$ is locally cartesian closed and cartesian monoidal and $C=V$.

However, the converse is false in general. Counterexamples can be found in this mathoverflow discussion (the discussion is only about cartesian closed enriched categories, but the counterexamples given are in fact locally cartesian closed, being indeed presheaf categories).

Examples

• Some kinds of type-theoretic model categories are required to be simplicially locally cartesian closed.

Related entries

• locally cartesian closed category

• cartesian closed enriched category

• monoidal enriched category

• locally cartesian closed model category

References

• Enriched cartesian closed categories on MO

• Simplicial cartesian closed categories on MO