A 2-category C has:
- A class of objects.
- For a pair of objects a category (not set) hom(x,y).
- Objects of hom(x,y) are morphisms of C.
- Morphisms of hom(x,y) are 2-morphisms of C.
hom(x,y) × hom(y,z) = hom(x,z)
Weak vs. Strong
If associativity and unit laws are upto equality then 2-category known as strong
If associativity and unit laws are upto isomorphism then 2-category known as weak
- weak 2-category known as bicategory.
- strong 2-category known as 2-category.
n-categories with only one object
|1||monoids||monoidal categories||monoidal 2-categories|
|2||commutative monoids||braided monoidal categories||braided monoidal 2-categories|
|3||"||symmetric monoidal categories||weakly involutory monoidal 2-categories|
|4||"||"||strongly involutory monoidal 2-categories|