Categories for the Working Mathematician
Encyclopedia
Categories for the Working Mathematician is a textbook in category theory
written by American
mathematician
Saunders Mac Lane
, who cofounded the subject together with Samuel Eilenberg
. It was first published in 1971, and is based on his lectures on the subject given at the University of Chicago
, the Australian National University
, Bowdoin College
, and Tulane University
. It is widely regarded as the premier introduction to the subject.
Chapters XI and XII were added in the 1998 second edition, the first in view of its importance in string theory
and quantum field theory
, and the second to address higher dimensional categories that have come into prominence.
Although it is the classic reference for category theory
, some of the terminology is not standard. In particular, Mac Lane attempted to settle an ambiguity in usage for the terms epimorphism
and monomorphism
by introducing the terms epic and monic, but the distinction is not in common use.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
written by American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
, who cofounded the subject together with Samuel Eilenberg
Samuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
. It was first published in 1971, and is based on his lectures on the subject given at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
, the Australian National University
Australian National University
The Australian National University is a teaching and research university located in the Australian capital, Canberra.As of 2009, the ANU employs 3,945 administrative staff who teach approximately 10,000 undergraduates, and 7,500 postgraduate students...
, Bowdoin College
Bowdoin College
Bowdoin College , founded in 1794, is an elite private liberal arts college located in the coastal Maine town of Brunswick, Maine. As of 2011, U.S. News and World Report ranks Bowdoin 6th among liberal arts colleges in the United States. At times, it was ranked as high as 4th in the country. It is...
, and Tulane University
Tulane University
Tulane University is a private, nonsectarian research university located in New Orleans, Louisiana, United States...
. It is widely regarded as the premier introduction to the subject.
Contents
The book has twelve chapters, which are:- Chapter I. CategoriesCategory (mathematics)In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
, FunctorFunctorIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
s, and Natural TransformationNatural transformationIn category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
s. - Chapter II. Constructions on Categories.
- Chapter III. Universals and LimitLimit (category theory)In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
s. - Chapter IV. AdjointAdjointIn mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type = .Specifically, adjoint may mean:...
s. - Chapter V. LimitsLimit (category theory)In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
. - Chapter VI. MonadMonad (category theory)In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations...
s and Algebras. - Chapter VII. MonoidMonoidal categoryIn mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
s. - Chapter VIII. Abelian CategoriesAbelian categoryIn mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
. - Chapter IX. Special Limits.
- Chapter X. Kan ExtensionKan extensionKan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M...
s. - Chapter XI. Symmetry and Braiding in Monoidal Categories
- Chapter XII. Structures in Categories.
Chapters XI and XII were added in the 1998 second edition, the first in view of its importance in string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
and quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, and the second to address higher dimensional categories that have come into prominence.
Although it is the classic reference for category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, some of the terminology is not standard. In particular, Mac Lane attempted to settle an ambiguity in usage for the terms epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
and monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
by introducing the terms epic and monic, but the distinction is not in common use.