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

If your perspective on nature has evolved up-to the point where you observe that life cannot be just a collection of random events but is structured and therefore naturally intelligent then category theory is a very interesting mathematical language that takes interest in structures and their inter-relationships. We shall look at some basic examples of categories today. 

Before trying to understand the examples, a word on the notion of object and arrow is due. An object carries a structure while an arrow or morphism refers to a correspondence between two objects carrying that common structure. Through the notion of correspondence we intuit that given an unknown object we can study it by setting up various possible correspondences with a known object. This is one of the most fundamental philosophy underlying the notion of arrows and their composition and therefore category theory itself. Here the mathematicians treat arrows and their composition more fundamentally than objects themselves. Or in other words an object of interest is studied relative to other objects with similar structures rather than an independent separate object. Such a philosophy is distinctly different (and therefore complementary as we shall through the blog) to that of set-theory on the basis of which our contemporary science models the object of study.

This shift in emphasis, where relationship plays a vital role at least as much as the objects themselves if not more, I believe is the pressing requirement of modern age. Today there are conflicts across religions, races, nations, fields of science. We cannot arrive at a unified treatment of issues because the models we have today don't sufficiently emphasise a relationship between objects and category theory is the mathematics of relationships. One such relationship viz. between cause and effect is emphasized in my thesis in the context of signal representation. 

Here are some images of the most widely refereed standard text-book of category theory by the mathematician who gave us the first definition of a category.

Take a look at a page of this book illustrating how definition is rigorously written. Without any pictures or visual cues in the book, it might appear as abstract nonsense but if you have read the posts on the definition then I hope you can see that these axioms actually carry physical meaning and they abstract away the essential structure of everyday life objects.

My point is that one must not fear such algebraic notations and always try to seek intuition and reason through visual figures. Now lets get back to the subject of this post. In illustrating the definition in earlier posts, I have used a single object in each category. They were chosen to give one of the interpretations of the arrows as transformations of an object as it adequately served our purpose to gain intuition for axioms. Also since these objects had a symmetry every arrow had an inverse. But now we shall generalize to the arrow as shown in the following figure where it intuitively signifies a general correspondence between two distinct domain and codomain objects.

A general arrow f:A->B in a category need not cover entire codomain although it is always well-defined on the entire domain. f(A) is termed as image or range of f and denoted as Im(f). Note that this is in general a small shaded oval inside the codomain and later we shall learn that this oval is a suboject of B.

Before looking at some standard examples of categories, let us consider a simple multiple object generalization of the category SET from earlier posts. Now we are considering a collection of four finite sets viz A={1,2,3,4}, B={x,y,z}, C={a,b,c,d} and D={m,n}. We form a category FINITE_SET consisting of four objects A,B,C and D. 

We can do so since these objects carry a certain common structure as we intuited earlier. Therefore what is the common (mathematical) structure on these objects ? It can be seen that each of the object is a set (on other words they have elements). Furthermore these elements are countably finite. Since we have recognised a structure underlying our objects we can set up various correspondences between them. Such correspondences or arrows are simply functions of sets one visual example of which we saw in the earlier post. Take a look at some sample arrows f,g and h in this category.

Will the permutations of sets be included in the arrows of this category? Yes because these are also functions which can be easily verified. Now one can verify all the axioms of the category as defined in the earlier posts. I have attached visual cues for your understanding.

Remember the identity arrow maps every element back to itself. Take a look at an example diagram of composition and unit law in the category. 

And finally an example of associativity axiom of the composition of three arrows in this category.

Although we have visually seen axioms being satisfied by taking some sample arrows, one can verify that any legitimate function will satisfy these axioms. In other words all possible functions between these sets form the collection of arrows in this category. We just saw a category FINITE_SET with four finite sets. 

In the next post we will further generalize the example of four finite sets to see how objects with a structure and structure preserving arrows form a category. As an example generalizing the example of collection of just four finite sets to include all possible finite sets and all possible functions between these finite sets we can form a category termed as  "FiniteSet" with objects as all finite sets and arrows as all functions between them.