1.1. From the Preface.

In presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with Otto Schilling. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with Samuel Eilenberg; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge their lively understanding of basic science. Both Reinhold Baer and Josef Schmid read and commented on my entire manuscript; their advice has led to many improvements. ... My wife, Dorothy, has cheerfully typed more versions of more chapters than she would like to count.

1.2. Review by: C Terence C Wall.
The Mathematical Gazette 49 (367) (1965), 105-106.

Homological algebra is a subject which grew up in the decade after the war as part of the process of formalising algebraic topology. It was at once recognised as providing also a powerful tool for pure algebra. The subject came of age with the book by H Cartan and S Eilenberg (1956) which brought several trial theories within a single framework. This book is difficult to read: there is a mass of new ideas, the relative importance of which is not always clear; the writing is rather technical; and the presentation depends on some familiarity with modules over algebras, which is not really essential to the subject. Mac Lane's book is a refreshing contrast. The author is at pains to introduce his concepts slowly, with discussion and examples to illustrate the points at issue. As a result, it is comparatively easy reading. He also has the advantage of ten years' work in the subject which have clarified the basic concepts and introduced some useful notations. His approach has the compensating defect of much increasing the length of the book: the basic concepts are worked over several times, with increasing degrees of abstraction-this reflects the current feeling that no single formulation is likely to be sufficiently general for all useful applications - and a prevailing impression is the diversification (not unification) of the subject. ... The book is likely to become a standard introduction to a subject which already has wide applications, and further work in which will probably be at a more abstract level.

1.3. Review by: Fred E J Linton.
Amer. Math. Monthly 71 (7) (1964), 818.

The process of assigning to a topological space its homology (or cohomology) groups is a two-step affair: first is produced a chain (or cochain) complex; from it, in turn, is calculated the (co)homology. Mac Lane's book treats, in the main, the second procedure - its tools (categories and functors, diagrams, spectral sequences), its algebraic applications (Tor, Ext, and the cohomology of monoids, groups, modules and algebras), its consequences (dimension theory, products) and its generalisation to abelian categories (derived functors, relative homological algebra, bar resolution). The topological aspects of homology theory, though not dominant in this book, are not totally ignored.

1.4. Review by: David A Buchsbaum.
Mathematical Reviews MR0156879 (28 #122).

Since the appearance of the first definitive book on homological algebra by Cartan and Eilenberg eight years ago [Homological algebra (1956)], the subject has had numerous applications and has undergone many changes. In this treatise on homology, the author not only incorporates his personal point of view in this field, but also includes many of these later developments both as additional material and as motivation for a good deal of the original subject matter of homological algebra.

2.1. From the Publisher.

This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance. This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach - emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s - was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of 'A Survey of Modern Algebra'. The present book presents the developments from that time to the first printing of this book.

2.2. Review by: Kenneth P Bogart.
Amer. Math. Monthly 78 (1) (1971), 92-93.

Professors Mac Lane and Birkhoff have once again set the pace and tone for instruction in abstract algebra for some time to come. I chose their book for my introductory course in abstract algebra for honours majors at Dartmouth. Since these students have a personal commitment to mathematics and frequently go on to graduate work, I felt strongly that their introduction to algebra should be couched in modern terminology and that their course should include relatively new algebraic concepts (e.g., categories and functors) that they were likely to see in other courses (e.g., topology). When I chose a text, I felt this one was the only one that satisfied these requirements. ... Although I strongly urge undergraduates who already know some algebra and first year graduate students to study this book carefully, I do not recommend it for an introductory course. Until we find a good way of teaching sophistication and sophisticated ideas in freshman calculus (I am not sure we even should), students will have too high a hurdle to jump in starting this book "cold." I found I could not cover the amount of material I wanted to (my students learned far less algebra than my regular majors in a one-term course from Fraleigh's book), that my students were overwhelmed by the formal style of the text, and that they did not develop much facility for using algebraic ideas and techniques.

2.3. Review by: Wilhelm Magnus.
Mathematics of Computation 22 (103) (1968), 693-694.

In spite of the similarity of the titles and the coincidence of the names (although not the sequence of names) of the authors, this is not a new edition of the 'Survey of Modern Algebra' (Macmillan Co., 1953) but a new book. The motivation for it is summarized in the first paragraph of the Preface: "Recent years have seen striking developments in the conceptual organization of mathematics. These developments use certain new concepts such as 'module,' 'category,' and 'morphism' which are algebraic in character and which indeed can be introduced naturally on the basis of elementary materials. The efficiency of these ideas suggests a fresh presentation of algebra." As in the Survey of Modern Algebra', the concepts and basic facts of the theory of sets, integers, groups, rings, fields, matrices, and vector spaces are introduced and proved; in addition, modules, lattices, multilinear algebra and other topics have their own chapters and are treated either in greater detail or as new subjects. But most of these chapters are used now also for the purpose of introducing and illustrating the concepts of "category," "functor," and "universal element" which, in the penultimate chapter on 'Categories and adjoint functors' become the main topic of the book.

2.4. Review by: Ancel C Mewborn.
Amer. Math. Monthly 74 (10) (1967), 1279.

It is natural to compare this book with the authors' A Survey of Modern Algebra since it is directed to the same audience. Most of the topics covered in the earlier book appear in some form in the present book; there is one notable exception: Galois Theory is omitted. ... Despite the large intersection of topics covered in Survey and in Algebra, the new book is not just an updating of the old. The treatment of many topics and the general tone differ greatly from that of Survey. The notation and terminology are "categorical," as are many proofs. Universal properties and duality are introduced in the first chapter and play an essential role throughout most of the book. Many concepts are defined in terms of, or are related to, universal mapping properties; and these properties are used when the concepts are applied.

2.5. Review by: Charles W Curtis.
Mathematical Reviews MR0214415 (35 #5266).

The concrete examples which underlie undergraduate algebra courses - integers, real and complex numbers, polynomials, vectors, matrices, determinants - are neither modern nor abstract. A textbook in which these objects are presented becomes a book on "modern algebra'' by the kind of scaffolding it erects to exhibit these objects. An earlier generation of mathematicians learned, for example in the authors' A survey of modern algebra (1941; 1953; 1965) that these objects are examples of groups, rings, fields, and vector spaces, and saw that the standard constructions in algebra often reduce to finding the right equivalence relation. It is refreshing to see in this book the standard material of undergraduate algebra presented systematically from a new point of view. This time the axiomatic systems and equivalence relations are there, but are governed in turn by the language of categories, functors, universal objects, and dualities.

2.6. From the Preface of the 2nd edition.

The treatment of many of the topics in the first edition has been simplified - and clarified - in this second edition. The material on universal constructions, formerly introduced at the end of the first chapter, has now been assembled in Chapter IV, at a point where there are at hand many more effective examples of these constructions. A great many points in the exposition have been clarified, for instance in a simpler construction of the integers, a more elementary description of polynomials, and a more direct treatment of dual spaces. The chapter on special fields now includes power series fields and a treatment of the p-adic numbers. There is a wholly new chapter on Galois theory; in exchange, the chapter on affine geometry has been dropped. New exercises have been added and some old slips have been excised.

2.7. Review of 3rd edition by: David Singmaster.
The Mathematical Gazette 75 (471) (1991), 121.

Birkhoff and Mac Lane's A Survey of Modern Algebra appeared in 1941 and made the ideas of modern algebra accessible to several generations of graduate and undergraduate students. The first edition of Algebra appeared in 1967 and was intended to incorporate the new ideas of algebra, such as module, tensor product, category and morphism. I well remember the surprise of opening the first edition and finding Chapter I was Sets, Functions, and Universal Elements, with commutative diagrams already on p 7 and the universal mapping property of the cartesian product on p 16. The logical sequence of the rest of the text was fairly natural: integers, groups, rings, fields, modules, vector spaces and linear algebra, structure of groups, lattices, categories, multilinear algebra - the novelty of the work was the use of the categorial viewpoint from the beginning. In the second edition of 1979, the authors reorganised their approach, changing the first chapter to Sets, Functions, and Integers and deferring the introduction of categorical ideas until Chapter IV. This seemed pedagogically sounder, as then the reader will have a supply of examples on which to base the generalisations. The chapters were somewhat reordered and the chapter on affine and projective spaces was replaced by one on Galois theory. In the present third edition, a number of misprints have been corrected, but the main text is otherwise identical to the second edition.

3.1. From the Publisher.

Category theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by mathematicians working in a variety of other fields of mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid - a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of generalized monoid. Chapters VI and VII explore this notion and its generalizations. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure (given by a tensor product) lead inter alia to the study of more convenient categories of topological spaces.

3.2. Review by: Alex Heller.
American Scientist 61 (3) (1973), 375-376.

This is a valuable book. Category theory has developed vigorously over a quarter of a century and has become an indispensable tool in many parts of mathematics. Now one of its cofounders presents us with a definitive synthesis designed to furnish to the working mathematician, i.e. the "consumer" of category theory, an understanding of the nature and the structure of this field. This is a far-from-trivial undertaking. The main ideas of category theory are interconnected in a remarkably complicated way. Thus the cycle of notions - monoid, category, monad, algebra - might be explicated starting. at almost any point and in almost any order. The same is true of the cycle-limit, universal construction, representable functor, adjoint functor. The relations between these cycles are more complex still. Professor Mac Lane has provided us not only with a clue through this labyrinth, but with a synopsis of its plan.

4.1. From the Introduction.

This book is intended to describe the practical and conceptual origins of Mathematics and the character of its development - not in historical terms, but in intrinsic terms. Thus we ask: What is the function of Mathematics and what is its form? In order to deal effectively with this question, we must first observe what Mathematics is. Hence the book starts with a survey of the basic parts of Mathematics, so that the intended general questions can be answered against the background of a careful assembly of the relevant evidence. In brief, a philosophy of Mathematics is not convincing unless it is founded on an examination of Mathematics itself. Wittgenstein (and other philosophers) have failed in this regard.

4.2. Review by: Penelope Maddy.
The Journal of Symbolic Logic 53 (2) (1988), 643-645.

The jacket blurb for Mac Lane's encyclopedic survey characterizes it as "a background for the philosophy of Mathematics." The qualification "background" should be emphasized because the bulk of the book, all but the introduction and the final chapter, contains little that is explicitly philosophical. Instead, the reader is treated to a masterful guided tour of the subject, from the natural, rational, ordinal, and cardinal numbers, Euclidean and non-Euclidean geometries, real and complex numbers, functions, transformations, groups, and the calculus, to linear algebra, differential geometry, mechanics, complex analysis, topology, set theory, logic, and category theory. These elegant and concise chapters clearly benefit from Mac Lane's own lifetime of productive engagement with his subjects. Appearances aside, however, this is not a book of popularized mathematics. The presentation is too dense for the uninitiated amateur; there is little concern for pedagogy, few concessions to the reader's fatigue or attention span, and little comic or other relief. History and anecdote are conspicuously absent, with a few notable exceptions, including a delightful ode on the Riemann conjecture, set to the tune of 'Sweet Betsy from Pike'.

4.3. Review by: Stephen L Bloom.
Studia Logica: An International Journal for Symbolic Logic 49 (3) (1990), 424-426.

This is a fascinating book. Its aim is to "capture in words a description of the form and function of Mathematics as a background for the Philosophy of mathematics". In brief, the plan of the book is to give a rather detailed summary of a good part of current topics in Mathematics, and use this material to evaluate various philosophical claims. ... It is claimed that the reader need have only "some acquaintance with Mathematics", but the level of sophistication needed to appreciate the discussion varies a great deal. However, for a mathematician, most of the discussions will be delightful, and informative. ... Although motivated by philosophical concerns, the book is best read as a source for excellent descriptions of actual mathematical fields of practice, of the roots of the major problems that motivate many areas of research, and of revelations of many interconnections among the various branches of Mathematics.

4.4. Review by: F Gareth Ashurst.
The Mathematical Gazette 71 (456) (1987), 176-177.

Professor Saunders Mac Lane is very familiar to most mathematicians for (among many other things) the two well known undergraduate textbooks on algebra of which he is the coauthor. The present book is a survey of mathematics expressly from the standpoint of structure and function. Deliberately the historical aspect of the subject, so common in many books which attempt a similar objective, has been played down. The nature of mathematical knowledge and the inter-relationships of various elements of the subject receive emphasis. The book originated in a series of lectures given by the author to postgraduate philosophy students specialising in the philosophy of mathematics. It is only to be expected that such students are both very able intellectually and not entirely unacquainted with mathematics. The book reflects this, and it is much more demanding on its readers than is usual with other works aimed at introducing a wide spectrum of mathematics to a non-specialist readership. ... Professor Mac Lane's depth of feeling for his subject comes out strongly through the whole of the book, one rather special instance of this is that Mathematics is always printed with a capital M.

4.5. Review by: Ivor Grattan-Guinness.
Mathematical Reviews MR0816347 (87g:00041)

The title of this book (in which 'Mathematics' and 'form' are separated by a colon, a comma or a space in its various appearances) promises a rather austere confinement to the more foundational and abstract aspects of the whole subject. By and large the auguries are preserved, for the emphases fall on the foundational aspects of abstract algebras, set theory, real- and complex-variable analysis, topology, geometry, and linear algebra: a somewhat lonely Chapter 9 on mechanics sits in the middle, with a strong preference shown for variational methods. ... The level is normally that around graduate level in mathematics. The account is, of course, very competent, but not especially exciting ...

5.1. From the Preface.

We dedicate this book to the memory of J Frank Adams. His clear insights have inspired many mathematicians, including both of us. In January 1989, when the first draft of our book had been completed, we heard the sad news of his untimely death. This has cast a shadow on our subsequent work. Our views of topos theory, as presented here, have been shaped by continued study, by conferences, and by many personal contacts with friends and colleagues-including especially O Bruno, P Freyd, J M E Hyland, P T Johnstone, A Joyal, A Kock, F W Lawvere, G E Reyes, R Solovay, R Swan, R W Thomason, M Tierney, and G C Wraith. Our presentation combines ideas and results from these people and from many others, but we have not endeavored to specify the various original sources.

5.2. From the Prologue.

A startling aspect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory. Indeed, a topos can be considered both as a "generalized space" and as a "generalized universe of sets". These different aspects arose independently around 1963: with A Grothendieck in his reformulation of sheaf theory for algebraic geometry, with F W Lawvere in his search for an axiomatization of the category of sets and that of "variable" sets, and with Paul Cohen in the use of forcing to construct new models of Zermelo-Fraenkel set theory. The study of cohomology for or generalized spaces led Grothendieck to define his notion of a topos. The cohomology was to be one with variable coefficients - for example, varying under the action of the fundamental group, as in N E Steenrod's work in algebraic topology, or more generally varying in a sheaf. The notion of a sheaf has its origins in the analytic continuation of functions, as initiated in the 19th century and then formulated rigorously in H Weyl's famous book on the "idea" of the Riemann surface.

5.3. Review by: Andrew M Pitts.
The Journal of Symbolic Logic 60 (1) (1995), 340-342.

What does the study of cohomology theories in algebraic geometry have in common with independence results in the logical and set-theoretical foundations of mathematics? One answer is that both have benefited from the use of sheaf theory. Roughly speaking, a sheaf over a space consists of a family of structures of the same kind parameterized by the elements of the space. Crucially, the parameterization has to respect the way in which the parts of the space fit together. The wide range of applications for sheaf theory is a result of the development of ever more general kinds of "space" for which there is a useful notion of sheaf, culminating in the notion of a topos. A sceptic might conclude that a topos must be such a general notion of space as to have rather shallow properties and be good for nothing in particular. The excellent introduction to topos theory contained in this book should convince such a sceptic that this is far from being the case. ... This is a well-conceived and carefully executed work which provides an invaluable introduction to its subject. Mac Lane's textbook 'Categories for the working mathematician' (Springer-Verlag, 1971) has been a mainstay for generations of students of category theory. I strongly suspect that his book with Moerdijk will achieve a similar status with students of topos theory

5.4. Review by: Michael Makkai.
Mathematical Reviews MR1300636 (96c:03119).

This book is a very detailed introduction to topos theory, aimed at readers with relatively little background in category theory. Recognizing the difficulty of understanding abstract machinery when it is presented without adequate motivation, the authors introduce the main concepts in a gradually increasing generality. ... The book is a very welcome addition to the literature. Its approach to exposition is the direct opposite to that of the "classic" in the field: P T Johnstone's Topos theory (1977). ... The present book, through its detailed discussions of points dealt with often in brief asides in Johnstone's book, through its many centrally important examples, and through its generally more friendly character will be a great help in the learning of the subject. ... I feel that the present book would have been enhanced by a more carefully documented history. The results and proofs are rarely attributed, and historical remarks are essentially confined to the prologue and the epilogue.

6.1. From the Publisher.

Saunders Mac Lane was an extraordinary mathematician, a dedicated teacher, and a good citizen who cared deeply about the values of science and education. In his autobiography, he gives us a glimpse of his "life and times," mixing the highly personal with professional observations. His recollections bring to life a century of extraordinary accomplishments and tragedies that inspire and educate. Saunders Mac Lane's life covers nearly a century of mathematical developments. During the earlier part of the twentieth century, he participated in the exciting happenings in Göttingen - the Mecca of mathematics. He studied under David Hilbert, Hermann Weyl, and Paul Bernays and witnessed the collapse of a great tradition under the political pressure of a brutal dictatorship. Later, he contributed to the more abstract and general mathematical viewpoints developed in the twentieth century. Perhaps the most outstanding accomplishment during his long and extraordinary career was the development of the concept of categories, together with Samuel Eilenberg, and the creation of a theory that has broad applications in different areas of mathematics, in particular topology and foundations. He was also a keen observer and active participant in the social and political events. As a member and vice president of the National Academy of Science and an advisor to the Administration, he exerted considerable influence on science and education policies in the post-war period. Mac Lane's autobiography takes the reader on a journey through the most important milestones of the mathematical world in the twentieth century.

6.2. Review by: Della D Fenster.
Amer. Math. Monthly 113 (10) (2006), 947-951.

If you had the good fortune to meet Saunders Mac Lane in your lifetime, his autobiography will give you a chance to have another conversation with him. If you never had the opportunity to meet Saunders Mac Lane in his lifetime, don't worry, his autobiography almost gives you a second chance. Three hundred and fifty pages come close to capturing his enormous personality and his enormous life. Only a book jacket of orange and brown plaid might have further enhanced this representation of the vibrant Mac Lane. A reader who casts a cursory glance at the table of contents will immediately recognize a man who did not wait for opportunities to find him. Quite the opposite: Mac Lane's autobiography stands as a final testimony to a man who embodied the Latin phrase 'Carpe diem'. ... The fifteen-part book offers something of a chronological overview of Mac Lane's life. Many of the chapters necessarily overlap since Mac Lane lived life to the fullest at home, in the classroom, on committees, as chair of his department, as an ambassador for mathematics, and in pursuit of new mathematics. His autobiography suggests that he tackled the demands of administrators, committee assignments, and students during the day and retreated to his study in the evening. Reflecting this same spirit, perhaps, Mac Lane devotes the first fourteen parts of his autobiography to the daily life of a mathematician and sets the final part aside for "Contemplating," the crown jewel of life for any member of what he calls the Academy. He could not have written anything other than a mathematical autobiography, since mathematics was Mac Lane and Mac Lane was mathematics.

6.3. Review by: Gerry Leversha.
The Mathematical Gazette 91 (522) (2007), 571-572.

For me, as for many other undergraduates taking their first steps in university mathematics, Saunders Mac Lane became familiar as the co-author, with Garrett Birkhoff, of 'A Survey of Modern Algebra'. This key text, published in 1941, had the explicit aim of persuading students of the value of the modern abstract approach. But this publication, relatively early in his academic life, was only one aspect of Mac Lane's multifaceted career as teacher, lecturer, researcher and advocate for mathematics. He knew, learnt from, taught and collaborated with many of the most celebrated figures in twentieth century mathematics, and worked in many different academic institutions both in the United States and Europe. This autobiography, written in his nineties, is a remarkable document. The first couple of chapters, dealing with the author's childhood, are a little uninspiring, probably because he cannot recall many details of his early life. There is even some confusion over the way in which the family name was changed from McLean to MacLane, although he does explain that the space in the final version was occasioned by his wife's inability to type it otherwise. Why should this be? - it might be interesting to know! To me, it seems that his autobiography only really begins when he discovers his vocation for mathematics, a fact he acknowledges in the chosen title. [Note by EFR. 'McLean' is pronounced 'Maclane' in Scotland. I suggest the name change was to get Americans to pronounce it correctly.]

6.4. Review by: Peter Johnstone.
Mathematical Reviews MR2141000 (2006b:01010),

Saunders Mac Lane died on 14 April 2005 at the age of 95, some six weeks before the publication of this book. Whilst it is of course a matter for deep sadness that he did not live to see his autobiography in print, the mathematical community can rejoice at the fact that he lived long enough to set the whole story down, with the encouragement in particular of his second wife Osa and his publisher Klaus Peters. For this is a splendid book, which will be read with great interest not only by those (such as the reviewer) who had the good fortune to know Mac Lane personally, but also by the much wider community of mathematicians whose lives have been influenced by his contributions to our subject. ... Throughout the book, the author's style is simple and direct, as one would expect. Sometimes it verges on the abrupt.