category object in an (∞,1)-category, groupoid object
equivalences in/of $(\infty,1)$-categories
The notion of n-category formulated internal to an ambient (∞,1)-category is supposed to externally be a kind of (∞,n)-category.
For $n =1$ this is a category object in an (∞,1)-category.
A model along these lines for general with the simplex category generalized to the Theta_n category and the ambient $(\infty,1)$-category being ∞Grpd are Theta_n spaces.
Another model is that of n-fold complete Segal space objects.
For the moment see at category object in an (infinity,1)-category – n-Categories