Tom Leinster

N = 0 : 0-Cat

• => Set, 0-morphism = function

N= 1: 1-Cat

• => Cat, 1-morphism = functor

N= 2: 2-Cat

• => 2-morphism = Natural Transformation

$latex text {f, g : 1-morphism }$

$latex alpha :: beta text { Natural Transformations : 2-morphism }$

Definition of n-Category:

Composition:

0-Cat : Set
1-Cat : Cat

Examples of n-Categories:

• Manifold
• Top (Topological Space) : 2-morphism = homotopy

Ref:

Best Technical Category Theory Book (2016) by Tom Leinster (Cambridge Press): “Basic Category Theory”

(Free downloadfrom arxiv)