Multiple categories

Let $T$ be a theory and let $C$ be a category. If $T$ is modelable in $T$, then there exists a category of models $T(C)$ in $C$. If $T$ is again modelable in $T(C)$ then we can get a category of double models $T(T(C))$. Repeating this again produces a category of triple models $T(T(T(C)))$. This iterated internalisation process produces the $n$-fold models of $T$ in $C$. This can be used to study a variety of different of theories in categories.

For example, we can repeatedly internalise the theory of preorders to categories to get $n$-fold preorders. However, the main things we will want to repeatedly internalise to categories are categories themselves. The iterated internalisations of categories in categories are $n$-fold categories. The double and triple categories we have dealt with previously are special types of $n$-fold categories. We call all $n$-fold categories with $n \gt 3$ multiple categories.

The morphisms of $n$-fold categories are $n$-fold functors. For example, the morphisms of single categories are simply single functors. The morphisms of double categories are double functors, and the morphisms of triple categories are triple functors. By generalisation, a morphism of multiple categories is a multiple functor. $n$-fold categories and $n$-fold functors together make up the categories of $n$-fold categories.

• Single categories
• Double categories
• Triple categories
• Multiple categories

References:
N-fold category