Dear Urs, You wrote:
The term "(infinity,1)-category" is not so much meant as an alternative for "quasi-category", but as a intentionally less specific term that subsumes concepts that are different from, but equivalent to, quasi-categories. Such as Kan-complex-enriched categories or complete Segal spaces, or algebraic quasi-categories, or categories with weak equivalences, or...
When doing abstract higher category theory it is useful to be able to speak, for instance, of the (infinity,1)-category of all small infinity-groupoids and its abstract properties, without having to specifically fix a concrete model in terms of which this entity may be brought to paper.
I agree that the terminology (infinity,1)-terminology can be useful. Can I point out that Lurie is calling a quasi-category an infinity-category? There is a clash of terminology. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]