Dear Toby, Looping and delooping operations can be applied to spaces and to maps between spaces. We should use a similar terminology for spaces and maps. For example: E-n space <--> E-n map Also for (higher) categories and functors. monoidal category <---> monoidal functor braided monoidal category <----> braided monoidal functor 2-braided monoidal category <--> 2-braided monoidal functor 3-braided monoidal category <--> 3-braided monoidal functor ...... ...... ...... symmetric monoidal category <--> symmetric monoidal functor A (n+1)-braided monoidal n-category is symmetric by the stabilisation hypothesis. I believe that a (n+1)-braided monoidal functor between (n+1)-braided monoidal n-categories is symmetric. Is this part of the official stabilisation hypothesis? Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]