|On a previous page we looked at ways to construct categories from existing categories. On this page we look at how objects in existing categories can become arrows in a new category.
A specific case of arrow categories is comma categories and a more specific case is slice categories. We can then further generalise to pullbacks.
In some circumstances we will see that certain universal properties are conserved.
|The arrow category gives us a way to convert objects into arrows.
|f: A -> X
|Objects in this arrow category are arrows between two categories. To specify this completely we need a triple <A,X,f> consisting of the two objects and the morphism between them.
|where 's' is an arrow in the source and 't' is an arrow in the target.
where the above diagram commutes, that is:
g•s = t •f
The comma category is like an arrow catagory but the source and target may be specified as functors from other categories. See page here.