An object of an -category is -truncated if is a -category for every object . Here (although the notion makes sense more generally) and , but neither need be directed (see n-prefix). For example:
In an -category every object is -truncated.
In a 1-category (or, actually, in any -category) the (-1)-truncated objects are the subterminals, and a terminal object is the only (-2)-truncated object.
In a 2-category, the (1,0)-truncated objects are the groupoidal ones, the (0,1)-truncated objects are the posetal ones, and the 0-truncated objects are the discrete ones.
We write for the full sub--category of spanned by the -truncated objects, which is a -category. If an object has a reflection into , we call this reflection the -truncation of and write it as . See truncation in an exact 2-category for ways to construct such truncations.
Revised on February 17, 2009 17:33:11
by Mike Shulman