An Elementary Overview of Mathematical Structures

There is much that is appealing about making explicit the categorical connections between different mathematical subjects.  To connect what we know about functions of sets to what we know about homomorphisms of groups to what we know about continuous functions of topological spaces and so forth is the epitome of pattern recognition. And that is the premise behind this book: to invite readers to relate various structures across algebra and topology via category theory. 

 

An Elementary Overview of Mathematical Structures is designed as an introduction to abstract algebra, an introduction to topology, followed by an introduction to the connective tissue of category theory.  This is not meant to provide a full first course in either algebra or topology; rather, the approach is to introduce structures that can be understood in terms of universal properties. The chapters on algebra start by discussing fields and vector spaces, then move on to abelian groups.  Groups more generally, as well as semigroups and monoids, appear shortly after that, followed by rings and more discussion of fields.  The chapters on topology begin with continuity in Euclidean space rather than with a definition of topological space. After that discussion, and the introduction of abstract topological space, the exposition includes quotients and products and sums.  There is also a discussion of countability and separation properties and limits before turning attention to connectedness, compactness etc.

 

For an undergraduate student new to either abstract algebra and point-set topology, this book presents an intense introduction.  Further, readers will be required to put the details behind some of the examples.  However, for an advanced reader, this would be a nice volume for self-study.

 

This book would have benefitted from more careful editing. In particular, the font size changes regularly throughout, but not for any reason I was able to discern.  Likewise, some sections are specifically labeled “Exercises and Complements” while some exercises appear at the end of other sections.  Some of the examples are meant to be exercises (as solutions appear for them in the last section); this is fine for a sophisticated reader who will verify on his or her own, but less optimal for students who expect a more explicit directive, especially when the text itself is phrased to indicate that the verification of the example has already been established.   

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Michele Intermont is an Associate Professor at Kalamazoo College.  Her research interests are in algebraic topology.