After beginning a book and reading for a little bit, I always turn to the introduction to see how the author envisions his/her book and whether or not that is squaring with my own impressions. If it is, I continue on, and if it's not, I begin again, looking to explain why my initial impression is not the same as the author's intention.

In this instance, Dowd states that his text is meant to serve as a introduction to abstract algebra for advanced undergraduates. But he doesn't stop there. Dowd comments:

There are many introductions to abstract algebra, so a new one should have some distinguishing characteristics. The main distinguishing characteristics of this text are the coverage of universal algebra and category theory, and their use in the introductory presentation of topics in algebra; and the coverage of various topics outside the main line of classical abstract algebra…

Indeed! This is not for your typical Abstract Algebra course. After a few preliminaries, the book starts with an exposition of universal algebra. Already, the third theorem of the book reads:

*If C is a non-empty collection of substructures of S, then* ∩* C is a substructure of S.*

Fifty pages into the 400 page manuscript, is a section entitled "Modules," which precedes the section on fields.

In many ways, this is a refreshing book. The perspective it brings is one I've come to advocate myself — that universal properties provide a useful set of organizing principles, giving both structure and flexibility to one's study. I'm impressed, too, with the topics covered. From sections on algebraic geometry and algebraic number theory to model theory and computability to homology. But in my opinion, for the framework of category theory to be meaningful, one should have already grappled with structures in a particular category. Then, in coming to similar structures in a different category, the idea of universality is more appealing.

So, the bottom line for me on *Introductory Algebra, Topology, and Category Theory* is that while I agree with Dowd's assessment that his book offers something different, and while I applaud what it offers, this isn't the text I would choose with which to introduce someone to abstract algebra. I will, however, freely market it to students who have completed a first abstract algebra course, and who are truly interested in mathematics. There is a lot of good stuff in here, much of which does not appear in the standard undergraduate courses, and it is written at a level that an undergraduate student can understand.

Michele Intermont teaches at Kalamazoo College.