Bergman, An Invitation to General Algebra and Universal Constructions

george-bergman's text for Math 245 at uc-berkeley — the graduate course in general (universal) algebra — in the version dated 2 April 2015, 551 PDF pages, held in raw/. Published by springer; the author keeps this file on his Berkeley page, which is how the corpus has it. The notes "began around 1971, as mimeographed outlines of my lectures" and were revised on every teaching since: forty-four years of one course, which makes this the longest-maintained text the spoke holds.

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What it covers

Ten chapters in three parts, and the shape is the argument. Part I spends 100 pages on examples before any general theory — free groups built three different ways (from terms, as subgroups of large direct products, and as groups of words), then a "Cook's tour" through presentations, abelianization, the Burnside problem, products and coproducts, tensor products, monoids, rings, Boolean algebras, and out into topology for the Stone–Čech compactification and universal covering spaces.

Only then does Part II build the tools: ordered sets, induction and the Axiom of Choice (including ordinals, Zorn's Lemma and a section called "Thoughts on set theory"); lattices, closure operators and Galois connections; categories and functors (~80pp, through natural transformations, functor categories and a sketch of enriched categories); universal constructions (Yoneda, adjoint functors, the p-adics, limits and colimits, existence theorems); and varieties of algebras — identities, derived operations, clones and clonal theories, Lawvere's Structure and Semantics.

Part III is one chapter on algebras, coalgebras and adjunctions, ending at Freyd's adjoint-functor criterion, which Bergman calls "the climax of the course."

Two things that make it unusual here

It names its own field's naming problem. §1.5, "The name of the game": the subject "has long been called Universal Algebra, but in recent decades, many workers in the field have come to dislike this term, feeling that 'it promises too much'." He uses General Algebra in the title and says the replacement has its own defect — "in some contexts, it may not be understood as referring to a specific area." No other text in the spoke opens by conceding that its subject's name is contested. See universal-algebra.

It states a prerequisite as a test, not a course list. "Anyone conversant with fewer than three ways of proving the existence of free groups has something to learn from Chapters 2-3." Formally he asks only for "an honors undergraduate algebra course or one semester of graduate algebra, plus a moderate level of mathematical sophistication" — but the operative bar is that sentence, and Chapter 3 then supplies exactly those three proofs. Compare grinberg-graph-theory, where a prerequisite list cut the syllabus; here the prerequisite is a self-assessment the book immediately satisfies.

The pedagogy, which is the sharpest part

Bergman gives seven pages to how the course is run and refuses to cut them from the published version, "since there are some nonstandard aspects to the way I run the course." The argument in §1.2 is not about rigour or content. It is about what class time is for:

"Why, for an hour at a time, should an instructor write notes on a blackboard and students copy them into their notebooks – often too busy with the copying to pay attention to the content – when this work could be done just as well by a photocopying machine?"

The notes exist to replace transcription, so that class can be spent on questions, discussion of exercises, and Socratic development. He then enforces the reading with a mechanism: a question a day (§1.3). Every student submits, in writing, at least one question about that day's reading, tagged urgent / important / unimportant / pro forma — and "if you give a 'pro forma' question, you must give the answer along with it!" Emailed an hour ahead, the answer gets worked into the lecture.

He is also blunt about the risk his own style creates. The notes contain a great deal of "handwaving (informal discussion)", and §1.4 says plainly: "I am worried that the amount of handwaving in these notes may lead some students to think handwaving is an acceptable substitute for proof." Handwaving "does not replace proofs" — it guides to them, or abbreviates one already seen, and where it abbreviates there is "a tacit challenge to you, to think through whether you can indeed fill in the steps."

Grading follows: one or two medium exercises per week, well done, is an A, with problems chosen by the student, partial credit "as long as you show you realize that they are partial", and joint submission allowed when contributions were roughly equal.

Exercises

629 exercises, interspersed in the text rather than collected at chapter ends, numbered in a separate system from theorems (m.n:k for exercises, m.n.k for everything else — a deliberate choice he justifies on findability grounds in §1.7). The back matter carries a List of Exercises with a telegraphic one-line description of each, so the reader can locate a problem by subject. No solutions. Some parts "may introduce an open question", and open questions are always marked as such.

Difficulty map, from the author

§1.7 tells other instructors where to slow down: "the hard material begins with Chapter 8", with the finer list being §§7.9–7.11, 8.3, 8.9–8.12, 9.9–9.10 and all of Chapter 10 — and, for students who have not met ordinals or categories before, the second halves of Chapters 5 and 7. The last two sections of Chapters 7, 8 and 9 are "sketchy (to varying degrees)" and are meant to be read for ideas. A text that publishes its own difficulty gradient is rare in this corpus.

Access and rights

© springer, and Bergman is explicit that keeping it free "will not be feasible for the final published version" — but he intends to keep the submitted version on his site, which is this file. So it sits with larsen-marx-mathematical-statistics and carroll-geometric-function-theory in the spoke's accessible-but-not-openly-licensed group, and unlike them the arrangement is stated by the author rather than inferred.

universal-algebra · category-theory · abstract-algebra · group-theory · george-bergman · uc-berkeley · springer · grinberg-graph-theory · judson-abstract-algebra · lal-algebra-1 · mathematics