ScalingStacks

Example 2.5. The following are some important examples of strict $n$-categories:

1. (2.5.1)

   The empty $n$-category $\emptyset$ is the empty set, regarded as an $n$-category. Later it will be convenient to write $\partial C_{0}\mathrel{\mathop{:}}=\emptyset$.

2. (2.5.2)

   The $0$-cell $C_{0}$ is the singleton set, viewed as a strict $n$-category. This is also the terminal strict $n$-category.

3. (2.5.3)

   The $1$-category $E$ is the “walking isomorphism,” that is, the unique contractible groupoid that contains exactly two objects.

4. (2.5.4)

   The $k$-cell $C_{k}$ is the strict $k$-category defined inductively as follows: the set of object of $C_{k}$ is the set $\{\bot,\top\}$, and one has

   $$\hom_{C_{k}}(x,y)\mathrel{\mathop{:}}=\begin{cases}C_{0}&\textrm{if }x=y;\\ C_{k-1}&\textrm{if }x=\bot\textrm{ and }y=\top;\\ \emptyset&\textrm{otherwise.}\end{cases}$$

   There is a unique composition law making this a strict $k$-category, and therefore a strict $n$-category for $n\geq k$.

5. (2.5.5)

   The $(k-1)$-category $\partial C_{k}\mathrel{\mathop{:}}=j_{k-1}C_{k}$ can be described as the $(k-1)$-category of “walking parallel $(k-1)$-morphisms”.

6. (2.5.6)

   A finite ordinal $S$ gives rise to a $1$-category $\Delta^{S}$, whose objects are elements of $S$ in which there is a unique morphism $s\to s^{\prime}$ if and only if $s\leq s^{\prime}$. The simplex category of nonempty finite ordinals will be denoted $\Delta$, as usual.