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

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

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, mathematical structure, structure, space, Mathematica ...

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 University of Chicago (UChicago, Chicago, or UChi) is a Private university, private research university in Chicago, Illinois, United States. Its main campus is in the Hyde Park, Chicago, Hyde Park neighborhood on Chicago's South Side, Chic ...

, the

Australian National University The Australian National University (ANU) is a public university, public research university and member of the Group of Eight (Australian universities), Group of Eight, located in Canberra, the capital of Australia. Its main campus in Acton, A ...

Bowdoin College Bowdoin College ( ) is a Private college, private liberal arts colleges in the United States, liberal arts college in Brunswick, Maine. It was chartered in 1794. The main Bowdoin campus is located near Casco Bay and the Androscoggin River. In a ...

, and

Tulane University The Tulane University of Louisiana (commonly referred to as Tulane University) is a private research university in New Orleans, Louisiana, United States. Founded as the Medical College of Louisiana in 1834 by a cohort of medical doctors, it b ...

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

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

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 Limits. :Chapter IV. Adjoints. :Chapter V. Limits. :Chapter VI. Monads and Algebras. :Chapter VII. Monoids. :Chapter VIII. Abelian Categories. :Chapter IX. Special Limits. :Chapter X. Kan Extensions. :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 and

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

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

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{{reflist 1971 non-fiction books Graduate Texts in Mathematics Mathematics textbooks Category theory

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