# nLab truncated object

category theory

## Applications

#### Higher category theory

higher category theory

# 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 $\left(n-1\right)$-category, a $k$-truncated object behaves like a $\mathrm{max}\left(k,n-1\right)$-category. More generally, a $\left(k,m\right)$-truncated object in an (n,r)-category is an object which behaves internally like a $\mathrm{max}\left(\left(k,m\right),\left(n-1,r-1\right)\right)$-category.

## Definition

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

## Examples

• In a 1-category:

• every object is $0$-truncated,
• the $\left(-1\right)$-truncated objects are the subterminal objects, and
• the $\left(-2\right)$-truncated objects are the terminal objects.
• In a 2-category:

• every object is $1$-truncated,
• the $0$-truncated objects are the discrete objects,
• the $\left(1,0\right)$-truncated objects are the groupoidal objects, and
• the $\left(0,1\right)$-truncated objects are the posetal objects.
• In an $\left(\infty ,1\right)$-category, the $k$-truncated objects (which are automatically $\left(k,0\right)$-truncated) are also called $k$-types. See n-truncated object of an (∞,1)-category.

## Properties

### Reflectivity

If the $\left(n,r\right)$-category has sufficient exactness properties, then the $\left(k,m\right)$-truncated objects form a reflective subcategory. More generally, in such a case there is a factorization system $\left(E,M\right)$ such that $M/1$ is the category of $\left(k,m\right)$-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, $\left(1,0\right)$-truncated, and $\left(0,1\right)$-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.

Revised on November 3, 2013 10:17:42 by Urs Schreiber (89.204.139.129)