n-category = (n,n)-category
n-groupoid = (n,0)-category
A -truncated object in an n-category is an object which “behaves internally like a -category”. More precisely, since an object of an -category can behave at most like an -category, a -truncated object behaves like a -category. More generally, a -truncated object in an (n,r)-category is an object which behaves internally like a -category.
Let be an -category, where and can range from to inclusive. An object is -truncated if for all objects , the -category is in fact a -category.
In a 1-category:
In a 2-category:
In an -category, the -truncated objects (which are automatically -truncated) are also called -types. See n-truncated object of an (∞,1)-category.
If the -category has sufficient exactness properties, then the -truncated objects form a reflective subcategory. More generally, in such a case there is a factorization system such that is the category of -truncated objects. (Note that this is not a reflective factorization system, but it is often a stable factorization system.) For example:
In a regular 2-category, the same holds true, where is the class of regular -epimorphisms (eso morphisms) and the class of -monomorphisms (ff morphisms). With additional exactness conditions, the categories of -truncated, -truncated, and -truncated objects are also reflective; see here.