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

diagram
  • Objects f:X->I
  • Morphisms <s,t>

Examples

fibration

Slice category

see page here

diagram
  • Objects f:X->I
  • Morphisms s
 

Co-slice category

see page here

diagram
  • Objects f:X->I
  • Morphisms t

Examples

substitution

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

Inside Functors

Page about functors here. A functor maps both elements and functions in some category.

In category theory we don't tend to look inside categories (such as the category of set). However I find it helps my intuition to attempt to derive properties of some categories (such as arrow category) from internal properties.

diagram

The mapping of functions must play well with the mapping of the elements.

g•F= F•f

or since g = F f :

(F f) • F= F•f

where • represents function composition.

diagram

Or we could have a contravarient functor:

(Ff)-1 • F = F •f

An alternative approach could be, rather than looking at an individual functor, we could try to look at all possible functions in terms of homsets:

diagram  

This can lead to structure in:

Co-fibration

diagram

Where:

  • x is a universal element.
  • P is a projection from E to B which we are trying to reverse.

When we try to reverse the projection P then:

  • a maps to a set of points s,s',s''...
  • b maps to a set of points t,t',t''...
  • f maps to g which is what? A function from one set to another.

 

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

Specific to this page here:

 

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