Category Theory Illustrated – Natural Transformations

Summary

Natural transformations I didn’t invent categories to study functors; I invented them to study natural transformations. — Saunders Mac Lane In this chapter, we will introduce the concept of a morphism between functors, or natural transformation. Understanding natural transformations will enable us to define category equality and some other advanced concepts. Natural transformations really are at the heart of category theory, however, their importance is not obvious at first. So, before introducing them, I like to talk, once more, about the body of knowledge that this heart maintains (I am good with metaphors… in principle). Equivalent categories Our first section aims to introduce natural transformation as a motivating example for creating a way to say that two categories are equal. But for that, we need to understand what equal categories are and should be. So, are you ready to hear about equivalent categories and natural transformations? Actually it is my opinion that you are not (no offence, they are just very hard!). So, we will take a longer route. I can put this next section anywhere in this book, and it would always be neither here nor there. But anyway, if you are studying math, you are probably interested in the nature of the universe. “What is the quintessential characteristic of all things in this world?” I hear you ask… Objects are overrated AKA Heraclitus was right! The world is the collection of facts, not of things. — Ludwig Wittgenstein What is the quintessential characteristic of all things in this world? Some 2500 years ago, the philosopher Parmenides gave an answer to this question, postulating that the nature of the universe is permanence, stasis. According to his view, what we perceive as processes/transformations/change is merely illusory appearances (“Whatever is is, and what is not cannot be”). He said that that things never really change, they only appear to change, or (another way to put it), only appearances change, but the essence does no...