# Maths - Category Theory - n-categories

### 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

 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 " " "

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Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.      The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.