homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
category object in an (∞,1)-category, groupoid object
An $n$-fold complete Segal space is a homotopy theory-version of an n-fold category: an $n$-fold category object internal to ∞Grpd hence an n-category object in an (∞,1)-category, hence an object in $Cat(Cat(\cdots Cat(\infty Grpd)))$. This is a model for an (∞,n)-category.
A complete Segal space is to be thought of as the nerve of a category which is homotopically enriched over Top: it is a simplicial object in Top, $X^\bullet : \Delta^{op} \to Top$ satisfying some conditions and thought of as a model for an $(\infty,1)$-category.
An $(\infty,n)$-category is in its essence the $(n-1)$-fold iteration of this process: recursively, it is a category which is homotopically enriched over $(\infty,n-1)$-categories.
This implies then in particular that an $(\infty,n)$-category in this sense is an $n$-fold simplicial topological space
which satisfies the condition of Segal spaces – te Segal condition (characterizing also nerves of categories) in each variable, in that all the squares
are homotopy pullbacks of $(n-1)$-fold Segal spaces.
In analogy of how it works for complete Segal spaces, the completness condition on an $n$-fold complete Segal space demands that the $(n-1)$-fold complete Segal space in degree zero is (under suitable identifications) the infinity-groupoid which is the core of the (infinity,n)-category which is being presented. Since the embedding of $\infty$-groupoids into ($n-1$)-fold complete Segal spaces is by adding lots of degeneracies, this means that the completeness condition on an $n$-fold complete Segal space involves lots of degeneracy conditions in degree 0.
(…)
The definition originates in the thesis
which however remains unpublished. It appears in print in section 12 of
The basic idea was being popularized and put to use in
A detailed discussion in the general context of internal categories in an (∞,1)-category is in section 1 of
For related references see at (∞,n)-category .