The Faces of Proteus

Look at the man on the cover of the book - you will see that he has no face. A geometric configuration is covered in its place, disintegrating each time we try to see a steady image in it. In this case, he reminds of one of the most mysterious characters of Greek mythology - the Old Man Proteus, the legend tells us that he could accept the appearance of every object, animal or plant ... If we try to draw his portrait, we would probably get the same: an abstract structure changing into a series of forms that seem to have nothing to do with each other.

The book you hold in your hands can also be seen as a portrait: its task is to show you the math as you have never seen it before. It tells about a new mathematical discipline born less than seventy years ago as a result of the collaboration between two of the most prominent mathematicians of our time - Saunders Mac Lane and Samuel Eilenberg. Its name is a category theory, and like the man in the painting who has no face, it does not have its own subject. Instead of being interested in certain types of objects such as numbers or geometric figures, it explores the mathematical structures themselves. Thus the category theory allows us to see the common between the different parts of mathematics, providing us with a universal language in which the results of any known formal theory can be explained.

Using hundreds of strictly defined definitions, detailed theorems, and detailed diagrams, this book will allow the reader to learn this language, which in the last quarter of a century finds dozens of applications in different scientific disciplines, from physics and mathematics through logic and computer science, to philosophy and social theory.