Synthesis of Modern Mathematics : Intuitive Category Theory (Definition I)

In this post, the goal is to intuitively understand the definition of a category through two very simple practical and concrete examples. 

 1.The first example is a four blade propeller, whose blades we shall label as 1,2,3 and 4. 

 2. Another example is a four element set A = {1,2,3,4}. 

Let us form two categories containing these objects. The first example of a propeller will give rise to a category we shall call PROP (the label reflects that its object is a propeller). The second example of a finite set will lead to a second category which we will call SET. Notation wise the categories are denoted using capital bold letters. 

The above figure is a visual depiction of the first axiom of a category. The first axiom says that every category consists of a collection of objects. In both categories that we have formed there is just a single object. But in general a category can have finitely or even infinitely many (collection of) objects. In fact mathematicians have also discovered categories with uncountable infinite collection of objects which they called a class. But we will not go into all such subtleties here. 

Now rotate the propeller clockwise by 90 degrees. This rotation of the propeller visually transforms the propeller shape again into same shape. Had we not labelled the blades we could not have distinguished that the propeller was rotated. This denotes an underlying structure on the propeller called symmetry. We shall label this rotation as 'r_90' and it is mathematically termed as an arrow (or a map or a morphism) of a category. In the figure below I demonstrate visually one sample arrow r_90 in the category PROP. You must note carefully that for defining an arrow of a category we require two objects which could be either same or different. The start object is called the domain and the end object is called the codomain. 

Now lets see another example of an arrow in the same category. Here we are rotating the propeller by 180 degrees. Such a rotation again preserves the symmetry of the object. 

The collection of arrows in the category PROP are all symmetry preserving transformations of the propeller. Can you count how many such rotations of the propeller are there which would preserve the its symmetry ? If we ponder, we have in all 8 rotations: 4 clockwise by 90, 180, 270 and 360 (or 0) degrees. Then we have 4 anti-clockwise rotations by 90, 180, 270 and 360 degrees. These rotations take the object in various states from which the object continues to look the same. Later you will realize that there are only 4 rotations corresponding to 4 unique states of the propeller object. 

Note that objects in a category carry some (mathematical) structure while arrows preserve the structure. The structure of the object in PROP is (visual) symmetry and certain precise rotations preserve this symmetry. Now let us try to define the arrows of second category SET. Here the object is a finite set. We can consider two structures on this object. We can consider a particular structure of being a finite set with exactly four elements. Else we could consider a general structure of just being a finite set (ignoring or forgetting that it has four elements).

If we consider the first structure then arrows will be the permutations of the set. This is because we have to preserve the structure of having finite elements and in particular exactly four elements. I demonstrate one such arrow f in the following figure.

Similarly if we forget the structure of having exactly four elements and only consider the structure that the object is a finite set then can you guess what could be the arrows ? In addition to all the permutation arrows we also have general mathematical functions as arrows as shown below. These arrows need not preserve the count of elements while mapping the elements. In other words the arrows can send multiple elements from the domain object to a single element in the codomain object.

We will only focus on permutation arrows (which can be inverted or reversed to go back to original state of object) and drop the general arrows as shown above for simplicity where we are preserving the count of elements in the set. 

Indeed we have visually depicted the second axiom of a category. It states that between any two objects of a category there is a collection of morphisms. Every map or arrow or morphism starts from an object called domain (of the arrow) and ends on an object called codomain (of the arrow). All objects with a given structure form the collection of objects of a category while the arrows are a collection which preserve that structure. 

In this post we learned two axioms (objects and arrows) that partly define a category. In the next post, we will learn few more axioms that will completely define a category.