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{: style="text-align: right"} "Try as you may,
{: style="text-align: right"} you just can't get away,
{: style="text-align: right"} from mathematics"
{: style="text-align: right" } Tom Lehrer About this book
The discipline of mathematics has always suffered from the fact that it is viewed as "science's workhorse". According to this view, mathematics is only "useful" as a means for making it easier for scientists and engineers to do their job making technological and and scientific advancements i.e. it is viewed as a tool for solving problems.
And although many people don't subsribe to this view specifically, we can see it encoded inside the structure of most mathematics text books - each chapter starts with an explanation of a concept, followed by some examples and then ends with a list of problems that this concept solves.
There is nothing wrong with this approach, but mathematics is so much more than solving problems. It was a the basis of a religious cult in ancient Grece (the Pythagoreans), it was seen by philosophers as means to understanding the laws which govern the universe. It was (and still is) a language, which can allow for people with different cultural backgrounds understand each other. And it was also art and a means of entertainment.
Category theory embodies all these aspects of mathematics. It's visual language is, I think a very good grounds to writing a book where all of them shine - a book that is based not on solving of problems, but on exploration of concepts and on seeking connections between them.
About category theory
The main reason I am interested in category theory is that it allows us to formalise some common concepts that we use in our daily (intellectual) lives. Much of our language is based on intuition as intuition is a very easy way to get your point across. However, that is part of the problem: sometimes intuition makes it too easy to communicate with someone, so easy that he might, in fact, understand things that you haven't actually said. For example, when I say that two things are equal, it would seem obvious to you what I mean, although it isn't obvious at all (how are they equal, at what context etc).
In such occasions we strive to use a more rigorous definition of what we are saying. This is a way to make it clearer and more understandable not only for our audience, but for ourselves. But providing such definition in natural language, which is designed to use intuition as a means of communication, is no easy task.
It is in these situations that people often resort to diagrams to explain their thoughts. Diagrams are ubiquitous in science and mathematics because they are an understandable way to communicate a formal concept clearly. Category theory formalises the concept of a diagram and their components - arrows and objects and creates a language for presenting all kinds of ideas. In this sense, category theory is a way to unify knowledge, both mathematical and scientific, and to bring various modes of thinking in common therms.
Summary
In this book we will visit various such modes of knowledge and along the way, we would see all other kinds of mathematical objects, viewed under the prism of categories.
We will start with set theory in chapter 1, which is the original way to formalize different mathematical concepts.
Chapter 2 we will do a (hopefully) gentle transition from sets to categories while showing how the two compare and (finally) introducing the definition of category theory.
In the next two chapters, 3 and 4 we would jump to two different branches of mathematics and will introduce their main means of abstraction, groups and orders, and we will see how do they connect to the core category-theoretic concepts that we introduced earlier.
Chapter 5 also follows the main formula of the previous two chapters, and it gets to the heart of the matter of why category theory is a universal language, by showing it's connection with the ancient discipline of logic. As in chapters 3 and 4 we start with a crash course of logic itself.
The connecting between all these different disciplines is examined in chapter 6, using one of the moist interesting category-theoretical concepts - the concept of a functor.
In chapter 7 we review another more interesting and more advanced categorical concept the concept of a natural transformation.
Acknowledgements
Thanks to my wife Dimitrina, who is taking after our daughter while I sit here and write my book.
Thanks to my high-school arts teacher, Mrs Georgieva who told me that I have some tallent, but I have to work.
Thanks to Prathyush Pramod who encouraged me to finish the book and is also helping me out with it.
And also to everyone else who submitted feedback and helped me fix some of the numerous errors that I made - knowing myself, I know that there are more.