A crucial point is whether the recipient of the enriching is cartesian or not. Note that fully internalising always must involve a cartesian aspect since one must diagonalize on the parametrizers of families of objects (at least) in order to explain eg natural transformations, even if the parametrizers for individual homs are not cartesian (eg linear or metric). One can envisage replacing individual "comma" categories by families of categories parametrized by (commutative) coalgebras, which seems just a way of constructing a cartesian category for the purpose, to which it may or may not be adequate. Symmetric monoidal categories in which the unit object is terminal seem to have a special role, but that may be illusory.(After all "any" smc is covered by one with that additional property) . Perhaps the affine modules ( see my paper "Grassmann's dialectics and category theory") constitute a good test case for proposed constuctions Bill Lawvere. ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* On Mon, 19 Oct 1998, Manuel Bullejos wrote:
Does any body know if comma categories have been defined in enriched contexts?
I have an idea of how they can be defined in some particular contexts, such as Cat-categories or Simplicial-categories, but I don't know if there is a general definition or even if a definition in the above two contexts can be found in the literature.
Thanks
Manuel Bullejos