Category Theory - Arrow, Comma and Slice 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.

Arrow Category

arrow category 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

comma category The comma category is a specialisation of the arrow category where the codomain is the same for all objects.
  Alternatively there is also a cocomma category where the domain is the same for all objects.
comma category The above diagram could be rewritten with the X shown in each object to make it look more like the arrow category.

The category of graphs is an example of a comma category, that is, a graph is isomorphic to a comma category. Arrows of graphs are pairs of functions mapping nodes to nodes (N -> N) and edges to edges (E -> E) , an object of this category it a triple (E,f: E -> N×N,N) where f maps each edge to a source and target node.

Then the comma category CoverX has,

The co-comma category XoverC has,

Objects

pairs(A,f)

Where:

  • 'A' is any object in C
  • 'f' is a mapping from A to X:
A  
over f
X  
pairs(A,f)
X  
over f
A  

Morphisms

The morphism f:
  s  
(A,f) -> (A',f')
Such that the diagram commutes
slice category
The morphism f:
  t  
(A,f) -> (A',f')
Such that the diagram commutes
coslice category

Terminal Objects in Comma

Initial Objects in Co-Comma

If C has a terminal object '1' exists then:

C/1 = C

If C has an initial object '0' exists then:

0/C = C

Products in Comma

CoProducts in Co-Comma

In terms of the above 'pairs' we can construct a product:
Take arbitrary elements:
A,B,D
and mappings of these to X
s : A->X, t : B->X,
μ : D->X
slice product

We can reduce the above diagram to:
slice product

 

Slice Category

This construction allows us to start with one category 'C' and generate a different category 'C/X' by fixing a given element 'X' in C. The elements in C/X are pairs (A,P) where A∈C and P is a morphism from A to X.

An example might be adding a distinguished point (the origin) to Euclidean space to give a vector space.

slice category overview

In the following we have also explained 'co-slice' the dual concept in the yellow boxes on the right.

Given a category C we can 'slice' it over some object X∈C which we fix in C.

 

 

Comma category is a more general case of the slice category (slice category is a specific case of comma category). this time we fix a functor f from category C to category D with an object X∈D.

So we fix f: C->D and X∈D

Then the slice category C/X has,

The coslice category X/C has,

Objects

pairs(c∈C,p:Fc->X)
Fc  
over P
X  
Where
pairs(c∈C,p:X->Fc)
X  
over P
Fc  

Morphisms

The morphism f: c->c'∈C
Such that the diagram commutes
comma category
The morphism f: c->c'∈C
Such that the diagram commutes
co comma category

We can generalise this further by not fixing X

overview comma category

Instead x is combined with the pair to give a triple.

Then the comma category FoverX has,

The co-comma category FoverC has,

Objects

triple(c∈C,x∈D,p)

Where:

  • x is any element in D (no longer fixed)
Fc  
over P
X  
triple(c∈C,x∈D,p)
X  
over P
Fc  

Morphisms

The morphisms are:
f: c->c'∈C
g: x->x'∈D
Such that the diagram commutes
comma category 2
 

Further generalisation, we add a third category 'E' like this:

comma category 3

Then the comma category FoverX has,

The co-comma category FoverC has,

Objects

quad(c∈C,e∈E,FoverX,p)
Fc  
over P
X  
Where x is any element in D (no longer fixed)
 

Morphisms

The morphisms are:
f: c->c'∈C
g: x->x'∈D
Such that the diagram commutes
comma category 2
 

 


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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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