### 2-categories

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.

Composition is:

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

from Categorification John C. Baez, James Dolan:

k | n=0 | n=1 | n=2 |

0 | sets | categories | 2-categories |

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 |

5 | " | " | " |