Let be a monoidal category with products and a -enriched category with products, in the enriched sense that we have a -natural isomorphisms of hom-objects in :
We say is -cartesian-closed if each -functor has a -enriched right adjoint.
If is -cartesian-closed, then its underlying ordinary category is 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, since the morphism of hom-objects induced by the evaluation morphism has invertible image in , hence is itself invertible if is conservative.
When is cartesian monoidal and : a cartesian closed category is automatically enriched-cartesian-closed over itself. In other words, the defining isomorphisms induce, by the Yoneda lemma, isomorphisms of exponential objects .
However, the converse is false in general. Counterexamples can be found in this mathoverflow discussion.
Last revised on May 31, 2023 at 12:25:13. See the history of this page for a list of all contributions to it.