## You are here

Homen-category

## Primary tabs

# n-category

###### Definition 0.1.

a small $n$-category , $\mathcal{C}_{n}$, is the $n$-th order category of (small) $n$-categories $n$-$\mathcal{C}at$ constructed by induction on $n$ in two main stages:

1. 2. define the category $(n+1)-\mathcal{C}at$ as the category of ($n$) categories enriched over the category $\mathcal{C}_{n}$. The construction is simplified by beginning with the definition of the 2-category.

1. define $\mathcal{C}at$ , or category $1-\mathcal{C}at$ as the category of small categories and functors;

2. 3. for all ‘$0$-cells’ $A$, $B$, consider the set $Hom_{{\mathcal{C}_{2}}}(A,B)$, or $\mathcal{C}_{2}(A,B)$, organized as a small category, whose $2$-morphisms, or ‘$1$-cells’, are defined as natural transformations called ‘$2$-cells’, $\eta:F\to G$ for any two ‘morphisms’ of $\mathcal{C}at$, with $F$ and $G$ being functors between the ‘$0$-cells’ $A$ and $B$, $F,G:A\to B$);

4. the 2-categorical composition is denoted as “$\bullet$” and is called the vertical composition;

5. a

*horizontal composition*, “$\circ$”, is defined for all triples of $0$-cells, $A$, $B$ and $C$ in $\mathcal{C}at$ as the functor $\circ:\mathcal{C}_{2}(B,C)\times\mathcal{C}_{2}(A,B)=\mathcal{C}_{2}(A,C)$; which is*associative*;6. the identities under horizontal composition are the identities of the $2$-cells of $1_{X}$ for any $X$ in $\mathcal{C}at$;

7. for any object $A$ in $\mathcal{C}at$ there is a functor from the one-object/one-arrow category $1$ (terminal object) to $\mathcal{C}_{2}(A,A)$.

8. repeat the last $(n-1)$ steps to define ‘3’-cells, …, to $n$-cells; the resulting structure is called an $n$-category, but it is in fact a metagraph, metacategory, or more generally, a $\S_{{n-1}}$-supercategory with

*$n$ composition laws*and it is also called more recently a*higher order category*or a*higher dimensional algebra*.

## Mathematics Subject Classification

18E05*no label found*18-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections