nLab
truncated object

Context

Category theory

Higher category theory

higher category theory

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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

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: