Read e-book online Categories for the Working Mathematician (2nd Edition) PDF

By Saunders Mac Lane

ISBN-10: 0387984038

ISBN-13: 9780387984032

An array of basic rules invaluable in a wide selection of fields. ranging from the principles, this e-book illuminates the innovations of type, functor, ordinary transformation, and duality. It then turns to adjoint functors, which offer an outline of common structures, an research of the representations of functors via units of morphisms, and a way of manipulating direct and inverse limits. those specific ideas are broadly illustrated within the final chapters, which come with many purposes of the fundamental life theorem for adjoint functors. the types of algebraic structures are made from convinced adjoint-like info and characterized by means of Beck's theorem. After contemplating quite a few purposes, the e-book maintains with the development and exploitation of Kan extensions. This moment version encompasses a variety of revisions and additions, together with new chapters on issues of lively curiosity: symmetric monoidal different types and braided monoidal different types, and the coherence theorems for them, in addition to 2-categories and the better dimensional different types that have lately come into prominence.

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Extra resources for Categories for the Working Mathematician (2nd Edition) (Graduate Texts in Mathematics, Volume 5)

Example text

This category Toph, which arises naturally in homotopy theory, shows that an arrow in a category need not be the same thing as a function. 8), and of Kan complexes. Set* will denote the category of small pointed sets (often called "based" sets). By a pointed set is meant a nonvoid set P with a selected element, written * or *p and called the "base point" of P. , which satisfies f(*p) = *Q. The pointed sets with these maps as morphisms constitute the category Set*. In this category the set {*} with just one point (the base point) is both an initial and a terminal object.

Under conjugation). A groupoid is said to be connected if there is an arrow joining any two of its objects. One may readily show that a connected groupoid is determined up to isomorphism by a group (one of the groups homG(x, x)) and by a set (the set of all objects). In this way, the fundamental groupoid n(X) of a path-connected space X is determined by the set of points in the space and a group homrr(x)(x, x) - the fundamental group of X. 21 Foundations Exercises 1. , dense subset of a topological space).

Is the set of all finite ordinals), (v) if I: a-+b is a surjective function with a E U and b c U, then b E U. These closure properties for U ensure that any of the standard operations of set theory applied to elements of U will always produce elements of U; in particular, WE U provides that U also contains all the usual sets of real numbers and related infinite sets. , on elements of U) while U itself and sets formed from U are to be used for the construction of the desired large categories. Now hold the universe U fixed, and call a set U E U a small set.

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Categories for the Working Mathematician (2nd Edition) (Graduate Texts in Mathematics, Volume 5) by Saunders Mac Lane

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