Category Theory - Arrow Categories

comma category overview 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>



Slice category

see page here

  • Objects f:X->I
  • Morphisms s

Co-slice category

see page here

  • Objects f:X->I
  • Morphisms t



Arrow Category

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 <X,I,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


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Book Shop - Further reading.

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.

flag flag flag flag flag flag 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.


Terminology and Notation

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