nLab
cubical truncation, skeleton, and co-skeleton

Contents

Truncation

We make use of the notation introduced in category of cubes and cubical set.

Notation

Let n0n \geq 0 be an integer. We denote by n\square_{n} the full sub-category of \square whose objects are I 0I^{0}, I 1I^{1}, \ldots, I nI^{n}.

Terminology

We refer to n\square_{n} as the nn-truncated category of cubes.

Notation

Let n0n \geq 0 be an integer. The inclusion functor i n: ni_{n} : \square_{n} \rightarrow \square canonically determines a functor i n *:Set opSet n opi_{n}^{*} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}}. We shall denote this functor by tr ntr_{n}.

Terminology

We refer to tr ntr_{n} as the nn-truncation functor.

Definition

Let n0n \geq 0 be an integer. The category of nn-truncated cubical sets is the free co-completion of n\square_{n}.

Remark

The free co-completion of a small category can be constructed as the category of presheaves presheaves of sets on this category. Thus we can also think of the category of nn-truncated cubical sets as the category of presheaves of sets on n\square_{n}.

Notation

We denote the category of nn-truncated cubical sets by Set n op\mathsf{Set}^{\square_{n}^{op}}.

Definition

An nn-truncated cubical set is an object of Set n op\mathsf{Set}^{\square_{n}^{op}}.

Remark

When we think of the category of nn-truncated cubical sets as the category of presheaves of sets on n\square_{n}, we consequently think of an nn-truncated cubical set as a presheaf of sets on n\square_{n}.

Definition

A morphism of nn-truncated cubical sets is an arrow of Set n op\mathsf{Set}^{\square_{n}^{op}}.

Terminology

Let XX be an nn-truncated cubical set. Let 0mn0 \leq m \geq n be an integer. By an mm-cube of XX, we shall mean an mm-cube of sk n(X)sk_{n}(X), where sk nsk_{n} is nn-skeleton functor defined in Notation .

Skeleton

Notation

Let n0n \geq 0 be an integer. By left Kan extension, the functor tr n:Set opSet n optr_{n} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}} admits a left adjoint i !:Set n opSet opi_{!} : \mathsf{Set}^{\square_{n}^{op}} \rightarrow \mathsf{Set}^{\square^{op}}. We shall denote this functor by sk nsk_{n}.

Terminology

We refer to sk nsk_{n} as the nn-skeleton functor.

Co-skeleton

Notation

Let n0n \geq 0 be an integer. By right Kan extension, the functor tr n:Set opSet n optr_{n} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}} admits a right adjoint i *:Set n opSet opi_{*} : \mathsf{Set}^{\square_{n}^{op}} \rightarrow \mathsf{Set}^{\square^{op}}. We shall denote this functor by cosk ncosk_{n}.

Terminology

We refer to cosk ncosk_{n} as the nn-skeleton functor.

Last revised on April 24, 2016 at 14:11:52. See the history of this page for a list of all contributions to it.