nLab locally cartesian closed enriched category

Locally cartesian closed enriched categories

Locally cartesian closed enriched categories


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

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

Relation to ordinary local cartesian closure

If CC is VV-locally-cartesian-closed, then its underlying ordinary category C 0C_0 is locally cartesian closed in the usual sense, since VV-enriched right adjoints have underlying ordinary right adjoints.

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

  • When V=SetV=Set, trivially.

  • More generally, whenever the underlying-set functor V(I,):VSetV(I,-) : V\to Set is conservative.

  • When VV is locally cartesian closed and cartesian monoidal and C=VC=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).



Last revised on May 31, 2023 at 12:24:49. See the history of this page for a list of all contributions to it.