# Category Theory - Arrow Categories

 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.

Here are some examples of comma categories. In these cases all the objects are in the same category (including the fixed object). The comma category generalises this by allowing the objects to come from other categories.

#### Arrow category

see page here

• Objects f:X->I
• Morphisms <s,t>

fibration

#### Slice category

see page here

• Objects f:X->I
• Morphisms s

#### Co-slice category

see page here

• Objects f:X->I
• Morphisms t

substitution

## Arrow Category

 The arrow category gives us a way to convert objects into arrows.
 objects: f: X -> I Objects in this arrow category are arrows between two categories. To specify this completely we need a triple consisting of the two objects and the morphism between them. morphisms: 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

## Comma Category

The comma category is like an arrow catagory but the source and target may be specified as functors from other categories. See page here.

## Related Categories

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