Michael Shulman
truncated object

An object AA of an nn-category KK is kk-truncated if K(X,A)K(X,A) is a kk-category for every object XKX\in K. Here n2n\le 2 (although the notion makes sense more generally) and kn1k\le n-1, but neither need be directed (see n-prefix). For example:

  • In an nn-category every object is (n1)(n-1)-truncated.
  • In a 1-category (or, actually, in any nn-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)trunc_k(K) for the full sub-nn-category of KK spanned by the kk-truncated objects, which is a min(n,k+1)min(n,k+1)-category. If an object AA has a reflection into trunc k(K)trunc_k(K), we call this reflection the kk-truncation of AA and write it as A kA_{\le 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)