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) 

1^{st }order  
2^{nd}order  
higher order 
This is often described in terms of ncells. In the diagrams below, the number of lines (Shafts) in the arrows indicates the order of the arrow.
0cell  1cell  2cell  3cell 

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,
Natural Transformations  
Monads  
falgebra  
fcoalgebra  
A given category 'C' with an endofunctor gives rise to a monad. By reversing the arrows (changing C to C^{op}) we can change the monad into a comonad. It may be that monad and comonad occur together. This happend when 3 adjoint functors are composed: LUR Examples:

Enriched Category
A category in which the homsets carry some extra structure, and that structure is preseved by function composition.