''Categories for the Working Mathematician'' (''CWM'') is a textbook in

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

written by American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...

, who cofounded the subject together with

Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...

. It was first published in 1971, and is based on his lectures on the subject given at the

University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...

, the

Australian National University The Australian National University (ANU) is a public research university located in Canberra, the capital of Australia. Its main campus in Acton encompasses seven teaching and research colleges, in addition to several national academies and ...

Bowdoin College Bowdoin College ( ) is a private liberal arts college in Brunswick, Maine. When Bowdoin was chartered in 1794, Maine was still a part of the Commonwealth of Massachusetts. The college offers 34 majors and 36 minors, as well as several joint eng ...

, and

Tulane University Tulane University, officially the Tulane University of Louisiana, is a private university, private research university in New Orleans, Louisiana. Founded as the Medical College of Louisiana in 1834 by seven young medical doctors, it turned into ...

. It is widely regarded as the premier introduction to the subject.

The book has twelve chapters, which are: :Chapter I.

Categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...

Functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

s, and

Natural Transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...

s. :Chapter II. Constructions on Categories. :Chapter III. Universals and

Limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

s. :Chapter IV. Adjoints. :Chapter V.

Limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

. :Chapter VI.

Monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', a ...

s and Algebras. :Chapter VII.

Monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

s. :Chapter VIII.

Abelian Categories In 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 prototypical example of an abelian category is the category of ...

. :Chapter IX. Special Limits. :Chapter X.

Kan Extension Kan extensions are Universal property, universal constructs in category theory, a branch of mathematics. They are closely related to Adjoint functors, adjoints, but are also related to Limit (category theory), limits and End (category theory), ends ...

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 In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...

and

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

, and the second to address higher-dimensional categories that have come into prominence. Although it is the classic reference for

, some of the terminology is not standard. In particular, Mac Lane attempted to settle an ambiguity in usage for the terms

epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...

and

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...

by introducing the terms ''epic'' and ''monic,'' but the distinction is not in common use.

References

Notes

{{reflist 1971 non-fiction books Mathematics books Monographs Category theory

Contents

References

Notes