homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
A $k$-truncated object in an n-category is an object which “behaves internally like a $k$-category”. More precisely, since an object of an $n$-category can behave at most like an $(n-1)$-category, a $k$-truncated object behaves like a $min(k,n-1)$-category. More generally, a $(k,m)$-truncated object in an (n,r)-category is an object which behaves internally like a $min((k,m),(n-1,r-1))$-category.
Let $C$ be an $(n,r)$-category, where $n$ and $r$ can range from $-2$ to $\infty$ inclusive. An object $x\in C$ is $(k,m)$-truncated if for all objects $a\in C$, the $(n-1,r-1)$-category $C(a,x)$ is in fact a $(k,m)$-category.
In a 1-category:
In a 2-category:
In an $(\infty,1)$-category, the $k$-truncated objects (which are automatically $(k,0)$-truncated) are also called $k$-types. See n-truncated object of an (∞,1)-category.
If the $(n,r)$-category has sufficient exactness properties, then the $(k,m)$-truncated objects form a reflective subcategory. More generally, in such a case there is a factorization system $(E,M)$ such that $M/1$ is the category of $(k,m)$-truncated objects. (Note that this is not a reflective factorization system, but it is often a stable factorization system.) For example:
In any category with a terminal object, the subcategory of terminal objects is reflective, and corresponds to the factorization system $E =$ all morphisms, $M=$ isomorphisms.
In a regular category, the category of subterminal objects is reflective, and corresponds to the factorization system $E=$ regular epimorphisms, $M=$ monomorphisms.
In a regular 2-category, the same holds true, where $E$ is the class of regular $1$-epimorphisms (eso morphisms) and $M$ the class of $1$-monomorphisms (ff morphisms). With additional exactness conditions, the categories of $0$-truncated, $(1,0)$-truncated, and $(0,1)$-truncated objects are also reflective; see here.
In an (∞,1)-topos, the $n$-truncated objects are reflective and we have the (n-connected, n-truncated) factorization system.