nLab cubical truncation, skeleton, and co-skeleton

Truncation

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

Notation

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

Terminology

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

Notation

Let $n \geq 0$ be an integer. The inclusion functor $i_{n} : \square_{n} \rightarrow \square$ canonically determines a functor $i_{n}^{*} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}}$. We shall denote this functor by $tr_{n}$.

Terminology

We refer to $tr_{n}$ as the $n$-truncation functor.

Definition

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

Remark

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

Notation

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

Definition

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

Remark

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

Definition

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

Terminology

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

Skeleton

Notation

Let $n \geq 0$ be an integer. By left Kan extension, the functor $tr_{n} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}}$ admits a left adjoint $i_{!} : \mathsf{Set}^{\square_{n}^{op}} \rightarrow \mathsf{Set}^{\square^{op}}$. We shall denote this functor by $sk_{n}$.

Terminology

We refer to $sk_{n}$ as the $n$-skeleton functor.

Co-skeleton

Notation

Let $n \geq 0$ be an integer. By right Kan extension, the functor $tr_{n} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}}$ admits a right adjoint $i_{*} : \mathsf{Set}^{\square_{n}^{op}} \rightarrow \mathsf{Set}^{\square^{op}}$. We shall denote this functor by $cosk_{n}$.

Terminology

We refer to $cosk_{n}$ as the $n$-coskeleton functor.