# Maths - Higher Order Category Theory

On these pages we discuss higher order categories.

We take first order category theory as consisting of objects and arrows between them.

There are various ways we can think about second order categories:

• By the addition of arrows between arrows (with appropriate axioms, these second level arrows are natural transformations).
• By allowing us to 'look inside' one layer of objects.

We can then go on to higher order objects by continuing the process (such as arrows between arrows between arrows and so on).

order between arrows inside objects
(in terms of components in D)
1st order
2ndorder
higher order

This is often described in terms of n-cells. Each higher order arrow has an extra shaft line.

0-cell 1-cell 2-cell 3-cell
object arrow arrow between arrows arrow between arrows between arrows

### Multicategories

 Another way to represent higher order categories is multicategories. In this case the arrows can have multiple inputs.

### Further topics

On the following pages we discuss some concepts of category theory, such as,

 A given category 'C' with an endofunctor gives rise to a monad. By reversing the arrows (changing C to Cop) we can change the monad into a co-monad. It may be that monad and co-monad occur together. This happend when 3 adjoint functors are composed: LUR Examples: propositional modal logic. open and closed subsets of a topological space.

#### Enriched Category

A category in which the hom-sets carry some extra structure, and that structure is preseved by function composition.

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.