ALGEBRA, Chapter 0 by Paolo Aluffi. (PDF)
From the introduction:
This text presents an introduction to algebra suitable for upper-level undergraduate or beginning graduate courses. While there is a very extensive offering of textbooks at this level, in my experience teaching this material I have invariably felt the need for a self-contained text that would start ‘from zero’ (in the sense of not assuming that the reader has had substantial previous exposure to the subject), but impart from the very beginning a rather modern, categorically-minded viewpoint, and aim at reaching a good level of depth. Many textbooks in algebra satisfy brilliantly some, but not all of these requirements. This book is my attempt at providing a working alternative.
There is a widespread perception that categories should be avoided at first blush, that the abstract language of categories should not be introduced until a student has toiled for a few semesters through example-driven illustrations of the nature of a subject like algebra. According to this viewpoint, categories are only tangentially relevant to the main topics covered in a beginning course, so they can simply be mentioned occasionally for the general edification of a reader, who will in time learn about them (by osmosis?). Paraphrasing a reviewer of a draft of the present text, ‘Discussions of categories at this level are the reason why God created appendices’.
It will be clear from a cursory glance at the table of contents that I think otherwise. In this text, categories are introduced around p. 20, after a scant reminder of the basic language of naive set theory, for the main purpose of providing a context for universal properties. These are in turn evoked constantly as basic definitions are introduced. The word ‘universal’ appears at least 100 times in the first three chapters.
If you are interested in a category theory based introduction to algebra, this may be the text for you. Suitable (according to the author) for use in a classroom or for self-study.
The ability to reason carefully, about what we imagine is known, should not be underestimated.
I first saw this in a tweet from Algebra Fact.