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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.
Arrow Category
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The arrow category gives us a way to convert objects into arrows. |
| objects: | 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. | ||
| morphisms: | <s,t> | 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
