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 $X\in K$. Here $n\le 2$ (although the notion makes sense more generally) and $k\le n-1$, but neither need be directed (see n-prefix). For example:

• In an $n$-category every object is $(n-1)$-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 $A_{\le k}$. See truncation in an exact 2-category for ways to construct such truncations.