Truncated objects

Idea

A $k$-truncated object in an n-category is an object which “behaves internally like a $k$-category”. More precisely, since an object of an $n$-category can behave at most like an $(n-1)$-category, a $k$-truncated object behaves like a $\max (k,n-1)$-category. More generally, a $(k,m)$-truncated object in an (n,r)-category is an object which behaves internally like a $\max ((k,m),(n-1,r-1))$-category.

Definition

Let $C$ be an $(n,r)$-category, where $n$ and $r$ can range from $-2$ to $\mathrm{\infty }$ inclusive. An object $x\in C$ is $(k,m)$-truncated if for all objects $a\in C$, the $(n-1,r-1)$-category $C(a,x)$ is in fact a $(k,m)$-category.

Examples

• In a 1-category:

  • every object is $0$-truncated,
  • the $(-1)$-truncated objects are the subterminal objects, and
  • the $(-2)$-truncated objects are the terminal objects.

• In a 2-category:

  • every object is $1$-truncated,
  • the $0$-truncated objects are the discrete objects,
  • the $(1,0)$-truncated objects are the groupoidal objects, and
  • the $(0,1)$-truncated objects are the posetal objects.

• In an $(\mathrm{\infty },1)$-category, the $k$-truncated objects (which are automatically $(k,0)$-truncated) are also called $k$-types. See n-truncated object of an (∞,1)-category.

Properties

Reflectivity

If the $(n,r)$-category has sufficient exactness properties, then the $(k,m)$-truncated objects form a reflective subcategory. More generally, in such a case there is a factorization system $(E,M)$ such that $M/1$ is the category of $(k,m)$-truncated objects. (Note that this is not a reflective factorization system, but it is often a stable factorization system.) For example:

• In any category with a terminal object, the subcategory of terminal objects is reflective, and corresponds to the factorization system $E=$ all morphisms, $M=$ isomorphisms.

• In a regular category, the category of subterminal objects is reflective, and corresponds to the factorization system $E=$ regular epimorphisms, $M=$ monomorphisms.

• In a regular 2-category, the same holds true, where $E$ is the class of regular $1$-epimorphisms (eso morphisms) and $M$ the class of $1$-monomorphisms (ff morphisms). With additional exactness conditions, the categories of $0$-truncated, $(1,0)$-truncated, and $(0,1)$-truncated objects are also reflective; see here.

• In an (∞,1)-topos, the $n$-truncated objects are reflective and we have the (n-connected, n-truncated) factorization system.