Let be a monoidal category and a -enriched category with pullbacks in the enriched sense. Then for any morphism in (the underlying ordinary category of) , there is a pullback -functor between the enriched slice categories, and each enriched slice category has -enriched products. We say is -locally-cartesian-closed if the following equivalent conditions hold:
If is -locally-cartesian-closed, then its underlying ordinary category is locally cartesian closed in the usual sense, since -enriched right adjoints have underlying ordinary right adjoints.
The converse is true in some cases, such as the following:
When , trivially.
More generally, whenever the underlying-set functor is conservative.
When is locally cartesian closed and cartesian monoidal and .
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).
Last revised on May 31, 2023 at 12:24:49. See the history of this page for a list of all contributions to it.