Dear Andre,
I agree that Lurie is using the infinity-n-category terminology. I am not questioning that. I am observing that he is calling a quasi-category an infinity-category. In his terminology, an infinity-category is a special kind of infinity-one-category.
I believe that the name infinity-category should apply to all "infinity" categories, inculding the infinity-one-category. No?
Yes, I entirely agree with that. In our discussion I did not promote Jacob Lurie's use of "oo-category" for "quasi-category", What I did and do promote is to use * "oo-groupoid" and "(oo,0)-category" as the term for the abstract concept which is equivalently realized by Kan complexes, simplicial groupoids, topological spaces, etc. and has special strict models by strict omega-groupoids, oo-fold groupoids etc. * "(oo,1)-category" as the term for the abstract concept which is equivalently realized by quasi-categories, Kan-complex enriched categories, complete Segal spaces, Segal categories, categories with weak equivalences, (oo,1)-theta spaces, etc. * "(oo,n)-category" as the term for the abstract concept which is equivalently realized by n-fold complete Segal spaces, (oo,n)-Theta spaces etc. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]