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

 In this post we will complete the definition of a category partially done in the last post. 

The third axiom says that for every object in a category there exists an identity arrow of the object. This intuitively says that for every object we can always do nothing and leave the object as it is or in other words transform the object in such a way that it leaves the object exactly in the same state from which we originally started. This translates to no rotation of the propeller (which is same as rotation by 0 degrees or rotation by 360 degrees). This rotation (or effectively no rotation) is the identity arrow of the propeller. Similarly in the case of second object, no permutation (which is same as the permutation in which we send every element to itself) is the identity arrow. Take a visual look to make sure you understand that identity arrow will always exists given an object because we have nothing in effect to do to the given object.

 Now if we ponder on the clockwise rotation by 180 degrees it can be discerned that it is basically two successive clockwise rotations by 90 degrees each. This is noting but composition axiom of a category in disguise. In the fist example observe that if we rotate the propeller by 90 degrees and further rotate it again by 90 degrees then those two rotations can be composed to form a new (composite) rotation which is nothing but rotation by 180 degrees. 

Let us look at another example of the propeller in which we rotate the blades first by 90 degrees and then by 180 degrees both clockwise then we have a composite arrow which is simply a rotation clockwise by 270 degrees.

Similarly in the case of object set S as shown the map f can be composed by map g and the effect of mapping the elements 1,2,3,4 by the composite map h is same as first mapping the elements by f then followed by mapping by g. We can verify this visually by using four different colours to track the elements as shown below.

We just intuitively demonstrated the composition axiom of a category. An arrow with a codomain object A can always be composed with another arrow having object A as its domain. Note that in the notation order is in the reverse way. That if f is followed by g we write as "g composed with f".

Rotation by 90 degrees followed by no rotation is same as no rotation first followed by rotation by 90 degrees which is identical to simple rotation by 90 degrees. This is the simple axiom of the unit law in a category.

The last axiom is pertaining to the composition of three arrows. We demonstrate it using the example of propeller. Observe that rotation by 90 degrees followed by rotation by 90 degrees which is again followed by rotation by 90 degrees is identical to rotation by 180 degrees followed by 90 degrees which is identical to rotation by 90 degrees followed by rotation by 180 degrees. This is a single transformation which is nothing but rotation by 270 degrees. It does not matter how we club the pair of transformations as seen in the figure. 

That is all there is to the definition of a category often dreaded as abstract nonsense yet so concrete and visually demonstrated in very basic objects of geometry and algebra.  For those who are well versed with the concept of a group will recognize that these examples are special cases of categories which are also groups (there is a single object propeller or a set and all the arrows have inverses, as an example anti-clockwise rotation by 90 degrees is the inverse of clockwise rotation by 90 degrees). The first category is nothing but a also cyclic group of order 4 while the second example is a permutation group. I have chosen these examples to emphasise the fact that historically category theory was born in trying to generalise group (as invariance and relativity was hot topic of research in those days)  or the concept that the study of invariant transformations characterize the structure of an object. This is thoroughly covered in the book  "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory" by Marquis, Jean-Pierre.

 If you loved following the first example then take a look at Nathan Carter's visual group theory for intuitive visual approach to group theory (which is special case of category theory). The visual approach here is motivated from that book. If you are more comfortable with the second example then refer the book "Conceptual Mathematics: A First Introduction to Categories" by Lawvere a very well-respected category theorist. 

All these axioms are summarized in the slide attached below:

In the next post we shall see several examples of categories pervading throughout mathematics and its applied fields including physics and engineering.