n-Categories

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 X and Y are isomorphic (X \cong Y) if there are functions f:X \to Y and g:Y \to X such that

    g \circ f = id_X, and f \circ g = id_Y.

  •   2: two categories X and Y are equivalent (X \simeq Y) if there are functors f:X \to Y and g:Y \to X such that

    g \circ f \cong id_X, and f \circ g \cong id_Y.

  •   …
  •   n: two n-categories X and Y are n-equivalent (X \simeq_n Y) if there are functors f:X \to Y and g:Y \to X such that

    g \circ f \simeq_{n-1} id_X, and f \circ g \simeq_{n-1} id_Y.

To be really pedantic, we could also have said

  •   0: two booleans X and Y are equal (X = Y) if there are equations f:X = Y and g:Y = X such that

    g \circ f = id_X, and f \circ g = id_Y,

where id_X is the equation X = X and similarly for Y.

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