Maths - Concrete Categories

concrete categories On these pages we look at concrete categorories. One definition of concrete categorories is a set together with some form of 'structure' such as functions and operations within the category.

Some of the simplest concrete categorories are:

Often in category theory we try to keep things as general as possible by working in terms of functor categories where we don't specify the internal structure

Finite Categories

Name Diagram Description
0   No objects
No arrows
1 category 1

One object
The only arrow is the arrow back to itself (identity arrow).

For instance: if the object is 'set' then the whole thing is a monoid.

2 category 2

Two objects

here are some examples:

two objects 1+1 that is the disjoint union of two single objects
category 2 This could be an index category
two objects This could be a graph
3 category 3

Three objects.
The arrow from first to last is implies by the composition law.

Here we avoid loops (exept identity) which would mean that something else is going on (such as isomorphism or adjunctions).

Categories with binary operation and identity element

  single objects multiple objects ->    
    composition law no composition law  
invertible Groups

Groupoids

all morphisms are isomorphisms

Graph  
non-invertible Monoids Categories Directed Graph  

Illustrating Category Constructions

In order to get an intuitive understanding of category theoretic constructions I find it helps to use examples from the simplest concrete categories to show what is going on internally, especially if it can be illustrated graphically. Of course, we are not really supposed to look inside categories, but I think its necessary to get an intuitive understanding.

Often sets are the simplest category to show whats going on.

If sets don't have enough structure the a pre-order (see this page) is often a good example to use (on this page I have tried to use preorders to show the idea of adjunctions). pre-orders are good because of the graphical nature of Hasse diagrams.

Pre-orders are also good because they are a specific (less general) case of categories in that, like categories, they are reflexive and transitive:

reflexivereflexive

a≤a

If a relationship is reflexive then there is always an arrow from an element of a set to itself. Although when drawing the Hasse diagram we may not draw it to avoid clutter.

transitivetransitive

if a≤b and b≤c then a≤c

This is like composing functions. That is: if the is an arrow from 'a' to 'b' and an arrow from 'b' to 'c', then we can attach the tip of the first arrow to the second arrow to get an arrow from 'a' to 'c'.

 


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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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