We have

- -1: a boolean T,
- 0: a set {T, F} of booleans,
- 1: a category Set of sets and functions,
- 2: a 2-category Cat of categories, functors, and natural transformations,
- …
- n: an (n+1)-category nCat of n-categories, functors, transformations, modifications, …, and n-cells.

In this hierarchy, a “0-category” is a set, a “(-1)-category” is a boolean, and the only “(-2)-category” is T. When we compare things in an n-category, we get this pattern:

- 0: two booleans are either equal or not,
- 1: two sets may be isomorphic without being equal,
- 2: two categories may be equivalent without being isomorphic,
- …
- n: two n-categories may be n-equivalent without being (n-1)-equivalent.

In particular,

- 1: two sets and are
**isomorphic**if there are functions and such that, and .

- 2: two categories X and Y are
**equivalent**if there are functors and such that, and .

- …
- n: two n-categories X and Y are
**n-equivalent**if there are functors and such that, and .

To be really pedantic, we could also have said

- 0: two booleans and are
**equal**if there are equations and such that, and ,

where is the equation and similarly for

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