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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
Goettingen---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.
This classic, written by two young instructors who became giants in
their field, has shaped the understanding of modern algebra for
generations of mathematicians and remains a valuable reference and
text for self study and college courses.
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
Goettingen---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.
This text introduces abstract algebra using familiar and concrete
examples that illustrate each new concept as it is presented. It
includes the coverage of such topics as the role of careful proof
in algebra; linear algebra as grounded in geometry; groups as
expressions of symmetry; subgroups and subsystems leading to
lattice theory; and more.
Covering a period up to 1971, this selection of Saunders Mac Lane's
most distinguished papers takes the reader on a journey through the
most important milestones of the mathematical world in the
twentieth century. Mac Lane was an extraordinary mathematician and
a dedicated teacher who cared earnestly about the values of science
and education. His life spanned nearly a century of mathematical
progress. In his earlier years, he participated in the exciting
developments in Goettingen. He studied under David Hilbert, Hermann
Weyl, and Paul Bernays. Later, he contributed to the more abstract
and general mathematical viewpoints which emerged in the twentieth
century. Perhaps the most outstanding accomplishment during his
long and extraordinary career was the development of the concept
and theory of categories, together with Samuel Eilenberg, which has
broad applications in different areas, in particular in topology
and the foundations of mathematics.
J. Richard Biichi is well known for his work in mathematical logic
and theoretical computer science. (He himself would have sharply
objected to the qualifier "theoretical," because he more or less
identified science and theory, using "theory" in a broader sense
and "science" in a narrower sense than usual.) We are happy to
present here this collection of his papers. I (DS)1 worked with
Biichi for many years, on and off, ever since I did my Ph.D. thesis
on his Sequential Calculus. His way was to travel locally, not
globally: When we met we would try some specific problem, but
rarely dis cussed research we had done or might do. After he died
in April 1984 I sifted through the manuscripts and notes left
behind and was dumbfounded to see what areas he had been in.
Essentially I knew about his work in finite au tomata, monadic
second-order theories, and computability. But here were at least
four layers on his writing desk, and evidently he had been working
on them all in parallel. I am sure that many people who knew Biichi
would tell an analogous story."
This book records my efforts over the past four years to capture in
words a description of the form and function of Mathematics, as a
background for the Philosophy of Mathematics. My efforts have been
encouraged by lec- tures that I have given at Heidelberg under the
auspices of the Alexander von Humboldt Stiftung, at the University
of Chicago, and at the University of Minnesota, the latter under
the auspices of the Institute for Mathematics and Its Applications.
Jean Benabou has carefully read the entire manuscript and has
offered incisive comments. George Glauberman, Car- los Kenig,
Christopher Mulvey, R. Narasimhan, and Dieter Puppe have provided
similar comments on chosen chapters. Fred Linton has pointed out
places requiring a more exact choice of wording. Many conversations
with George Mackey have given me important insights on the nature
of Mathematics. I have had similar help from Alfred Aeppli, John
Gray, Jay Goldman, Peter Johnstone, Bill Lawvere, and Roger Lyndon.
Over the years, I have profited from discussions of general issues
with my colleagues Felix Browder and Melvin Rothenberg. Ideas from
Tammo Tom Dieck, Albrecht Dold, Richard Lashof, and Ib Madsen have
assisted in my study of geometry. Jerry Bona and B. L. Foster have
helped with my examina- tion of mechanics. My observations about
logic have been subject to con- structive scrutiny by Gert Miiller,
Marian Boykan Pour-El, Ted Slaman, R. Voreadou, Volker Weispfennig,
and Hugh Woodin.
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 SCHIL LING. A further development of
homological ideas, with a view to their topological applications,
came in my long collaboration with SAMUEL ElLENBERG; 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. ANDERS KOCK and JACQUES RIGUET have read
the entire galley proof and caught many slips and obscurities.
Among the others whose sug gestions have served me well, I note
FRANK ADAMS, LOUIS AUSLANDER, WILFRED COCKCROFT, ALBRECHT DOLD,
GEOFFREY HORROCKS, FRIED RICH KASCH, JOHANN LEICHT, ARUNAS
LIULEVICIUS, JOHN MOORE, DIE TER PUPPE, JOSEPH YAO, and a number of
my current students at the University of Chicago - not to m ntion
the auditors of my lectures at Chicago, Heidelberg, Bonn,
Frankfurt, and Aarhus. My wife, DOROTHY, has cheerfully typed more
versions of more chapters than she would like to count. Messrs."
Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.
This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds (algebraic, analytic, etc.). Sheaves also appear in logic as carriers for models of set theory as well as for the semantics of other types of logic. Grothendieck introduced a topos as a category of sheaves for algebraic geometry. Subsequently, Lawvere and Tierney obtained elementary axioms for such (more general) categories. This introduction to topos theory begins with a number of illustrative examples that explain the origin of these ideas and then describes the sheafification process and the properties of an elementary topos. The applications to axiomatic set theory and the use in forcing (the Independence of the Continuum Hypothesis and of the Axiom of Choice) are then described. Geometric morphisms- like continuous maps of spaces and the construction of classifying topoi, for example those related to local rings and simplicial sets, next appear, followed by the use of locales (pointless spaces) and the construction of topoi related to geometric languages and logic. This is the first text to address all of these varied aspects of topos theory at the graduate student level.
An array of general ideas useful in a wide variety of fields.
Starting from the foundations, this book illuminates the concepts
of category, functor, natural transformation, and duality. It then
turns to adjoint functors, which provide a description of universal
constructions, an analysis of the representations of functors by
sets of morphisms, and a means of manipulating direct and inverse
limits. These categorical concepts are extensively illustrated in
the remaining chapters, which include many applications of the
basic existence theorem for adjoint functors. The categories of
algebraic systems are constructed from certain adjoint-like data
and characterised by Beck's theorem. After considering a variety of
applications, the book continues with the construction and
exploitation of Kan extensions. This second edition includes a
number of revisions and additions, including new chapters on topics
of active interest: symmetric monoidal categories and braided
monoidal categories, and the coherence theorems for them, as well
as 2-categories and the higher dimensional categories which have
recently come into prominence.
Department Of Mathematics, University Of Chicago, Autumn 1953.
Prepared With The Assistance Of A Grant By The Research
Corporation.
Department Of Mathematics, University Of Chicago, Autumn 1953.
Prepared With The Assistance Of A Grant By The Research
Corporation.
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