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 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.