Michael Shulman
truncated object

An object A of an n-category K is k-truncated if K(X,A) is a k-category for every object XK. Here n2 (although the notion makes sense more generally) and kn1, but neither need be directed (see n-prefix). For example:

  • In an n-category every object is (n1)-truncated.
  • In a 1-category (or, actually, in any n-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 trunc k(K) for the full sub-n-category of K spanned by the k-truncated objects, which is a min(n,k+1)-category. If an object A has a reflection into trunc k(K), we call this reflection the k-truncation of A and write it as A k. See truncation in an exact 2-category for ways to construct such truncations.

Revised on February 17, 2009 17:33:11 by Mike Shulman (75.3.140.11)