Author | Saunders Mac Lane |
---|---|

Country | United States |

Language | English |

Series | Graduate Texts in Mathematics; Vol. 5 |

Subject | Category theory |

Publisher | Springer Science+Business Media |

Publication date | 1971 |

Media type | |

Pages | 262 |

ISBN | 0-387-90036-5 |

OCLC | 488352436 |

512.55 | |

LC Class | LCC QA169.M33 |

* Categories for the Working Mathematician* (

The book has twelve chapters, which are:

- Chapter I. Categories, Functors, and Natural Transformations.
- 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, and the second to address higher-dimensional categories that have come into prominence.^{ [3] }

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.^{ [4] }

**Category theory** formalizes mathematical structure and its concepts in terms of a labeled directed graph called a *category*, whose nodes are called *objects*, and whose labelled directed edges are called *arrows*. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

In mathematics, a **category** is a collection of "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 objects are sets and whose arrows are functions.

In category theory, a branch of mathematics, an **initial object** of a category C is an object I in C such that for every object X in C, there exists precisely one morphism *I* → *X*.

In mathematics, an **algebraic structure** consists of a nonempty set *A*, a collection of operations on *A* of finite arity, and a finite set of identities, known as axioms, that these operations must satisfy.

In category theory, the **product** of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

In category theory, a branch of mathematics, **duality** is a correspondence between the properties of a category *C* and the dual properties of the opposite category *C*^{op}. Given a statement regarding the category *C*, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category *C*^{op}. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about *C*, then its dual statement is true about *C*^{op}. Also, if a statement is false about *C*, then its dual has to be false about *C*^{op}.

**Saunders Mac Lane** was an American mathematician who co-founded category theory with Samuel Eilenberg.

**Samuel Eilenberg** was a Polish-American mathematician who co-founded category theory and homological algebra.

In mathematics, **abstract nonsense**, **general abstract nonsense**, **generalized abstract nonsense**, and **general nonsense** are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, “abstract nonsense” may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself.

**Francis William Lawvere** is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.

**Categorical logic** is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970.

**Garrett Birkhoff** was an American mathematician. He is best known for his work in lattice theory.

In mathematics, a **pointed set** is an ordered pair where is a set and is an element of called the **base point**, also spelled **basepoint**.

In mathematics, specifically in category theory, an **exponential object** or **map object** is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories without adjoined products may still have an **exponential law**.

In mathematics, a **free Boolean algebra** is a Boolean algebra with a distinguished set of elements, called * generators*, such that:

- Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and
- The generators are as
*independent*as possible, in the sense that there are no relationships among them that do not hold in*every*Boolean algebra no matter*which*elements are chosen.

In algebra, given a ring *R*, the **category of left modules** over *R* is the category whose objects are all left modules over *R* and whose morphisms are all module homomorphisms between left *R*-modules. For example, when *R* is the ring of integers **Z**, it is the same thing as the category of abelian groups. The **category of right modules** is defined in a similar way.

**Izak (Ieke) Moerdijk** is a Dutch mathematician, currently working at Utrecht University, who in 2012 won the Spinoza prize.

* Introduction to Lattices and Order* is a mathematical textbook on order theory by Brian A. Davey and Hilary Priestley. It was published by the Cambridge University Press in their Cambridge Mathematical Textbooks series in 1990, with a second edition in 2002. The second edition is significantly different in its topics and organization, and was revised to incorporate recent developments in the area, especially in its applications to computer science. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.

- Mac Lane, Saunders (September 1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics.**5**(Second ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001.

- ↑ Leinster, Tom (2014).
*Basic Category Theory*. Cambridge University Press. p. 174. "The towering presence among category theory books is the classic one by one of its founders: Saunders Mac Lane's Categories for the Working Mathematician" - ↑ Awodey, Steve (2010).
*Category Theory*. Oxford University Press. p. iv. "Why write a new textbook on Category Theory, when we already have Mac Lane’s Categories for the Working Mathematician? Simply put, because Mac Lane’s book is for the working (and aspiring) mathematician. What is needed now, after 30 years of spreading into various other disciplines and places in the curriculum, is a book for everyone else." Awodey also dedicated the book to Saunders Mac Lane. - ↑ From the preface to the second edition.
- ↑ Bergman, George (1998).
*An Invitation to General Algebra and Universal Constructions*. Henry Helson. p. 179.

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