nLab truncated object

Redirected from "posetal object".
Truncated objects

Context

Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Truncated objects

Idea

A kk-truncated object in an n-category is an object which “behaves internally like a kk-category”. More precisely, since an object of an nn-category can behave at most like an (n1)(n-1)-category, a kk-truncated object behaves like a min(k,n1)min(k,n-1)-category. More generally, a (k,m)(k,m)-truncated object in an (n,r)-category is an object which behaves internally like a min((k,m),(n1,r1))min((k,m),(n-1,r-1))-category.

Definition

Let CC be an (n,r)(n,r)-category, where nn and rr can range from 2-2 to \infty inclusive. An object xCx\in C is (k,m)(k,m)-truncated if for all objects aCa\in C, the (n1,r1)(n-1,r-1)-category C(a,x)C(a,x) is in fact a (k,m)(k,m)-category.

Examples

  • In a 1-category:

  • In a 2-category:

    • every object is 11-truncated,
    • the 00-truncated objects are the discrete objects,
    • the (1,0)(1,0)-truncated objects are the groupoidal objects, and
    • the (0,1)(0,1)-truncated objects are the posetal objects.
  • In an (,1)(\infty,1)-category, the kk-truncated objects (which are automatically (k,0)(k,0)-truncated) are also called kk-types. See n-truncated object of an (∞,1)-category.

Properties

Reflectivity

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

Last revised on November 24, 2023 at 06:34:33. See the history of this page for a list of all contributions to it.