One of my goals for this semester is to work through Paolo Aluffi’s Algebra: Chapter 0, and see how far I can get. I’ve been meaning to properly learn abstract algebra for quite a while now; up to this point, my knowledge of algebra consists of bits and pieces that I’ve picked up here and there, to understand various applications: fundamental groups in algebraic topology, categories and functors, monads in Haskell, and theorems concerning rings, polynomials, and ideals, for an abstract linear algebra class.
For this reason, I’ve decided to go with Aluffi’s algebra book, which is supposed to provide a self-contained introduction to abstract algebra that is, in the author’s words, “suitable for upper-level undergraduate or beginning graduate courses.” One of the unique features of this book is that it uses category theory to interweave separate algebraic topics right from the start, rather than relegating a brief discussion of categories and functors to an appendix. I, for one, am a fan of category theory, especially when it helps to elucidate the connections between seemingly disparate topics in math.
Every undergraduate-level math text inevitably starts with a review of the basics: sets, functions, relations, and all that. Aluffi goes a step further, defining the basic language of categories that will be used throughout the book to define and unify concepts involving groups, rings, modules, and other algebraic objects. Since most of the topics in Chapter 1 simply reviewed material that I had seen before, I mostly skimmed over this chapter, and quickly glanced at most of the exercises to make sure that I knew at least the general idea of how to solve them.
Two things struck me as I read the chapter. The first is that Aluffi didn’t choose to number the chapters of his books starting from zero. The other thing that seemed a bit strange to me, at least at fist, was the omission of a proper definition of functors. The word “functor” is briefly mentioned towards the end of the chapter, in the section on universal properties, as being part of the underlying formalism used to make concepts surrounding universal properties rigorous. I would argue that, at this point, enough category-theoretic terminology has been introduced to at least define functors, and to briefly discuss diagrams, cones, limits, and all that. But I concede that this is not a major omission; at this point in the text this level of formalism is not exactly necessary, but it might be nice to have. Then again, this is a text on abstract algebra, not category theory. For a thorough discussion of these topics, several excellent books already exist, such as Steve Awodey’s excellent book, aptly titled Category Theory.
As I work through later chapters of this book, I may also post solutions to some of the exercises. Though I want to be a bit careful how much I post, since I’m sure that a fair number of professors use this book for their algebra classes, and I don’t want to spoil too many solutions (or mislead students with potentially wrong answers).
For now though, I think I’m ready to dive into the next chapter, which starts getting into the meat of the subject: groups!