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0 pointsauggierose2y ago0 comments

Yet its origin seems to be something quite concrete and practical.

For example, just recently I came across this text: "Foundations of Algebraic Topology", by Eilenberg and Steenrod. Its preamble is highly readable and engaging, see below. We have a topology and compute some algebraic structure from it. Sounds easy!

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The principal contribution of this book is an axiomatic approach to the part of algebraic topology called homology theory. It is the oldest and most extensively developed portion of algebraic topology, and may be regarded as the main body of the subject. The present axiomatization is the first which has been given. The dual theory of cohomology is likewise axiomatized. It is assumed that the reader is familiar with the basic concepts of algebra and of point set topology. No attempt is made to axiomatize these subjects. This has been done extensively in the literature. Our achievement is different in kind. Homology theory is a transition (or function) from topology to algebra. It is this transition which is axiomatized. Speaking roughly, a homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist's field of study. Its range is the field of study of the algebraist. Topological problems are converted into algebraic problems. In this respect, homology theory parallels analytic geometry. How ever, unlike analytic geometry, it is not reversible. The derived algebraic system represents only an aspect of the given topological system, and is usually much simpler. This has the advantage that the geometric problem is stripped of inessential features and replaced by a familiar type of problem which one can hope to solve. It has the disadvantage that some essential feature may be lost. In spite of this, the subject has proved its value by a great variety of successful applications. Our axioms are statements of the fundamental properties of this assignment of an algebraic system to a topological system. The axioms are categorical in the sense that two such assignments give isomorphic algebraic systems.

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jasomill2y ago

Eilenberg and Steenrod is great, but if you're not quite ready for algebraic topology, Mac Lane and Birkhoff's _Algebra_ [1] is an introductory abstract algebra text that makes well-motivated use of category theory throughout. From the preface:

…we hold that the general and abstract ideas needed should grow naturally from concrete instances. With this in view, it is fortunate that we do not need to begin with the general notion of a category. The most basic category is the category whose objects are all sets and whose morphisms are all functions (from one set to another); hence we can start Chapter I with sets — more accurately, with sets, functions, and the composition of functions — as the fundamental materials. On this background, Chapter II introduces the integers as the most basic example of an algebraic system. All the other categories which we need are quite "concrete" ones — each object A in the category is a set (with some structure), and each morphism from an object A to an object B in the category is a function (one which preserves the structure) on the set A to the set B. Hence we can give in Chapter I an easy, explicit definition of a "concrete" category, leaving the full treatment of the more general notion of a category to Chapter XV. In the same spirit, the idea fundamental to the notion of adjoint functor turns out to be the simple one of a "universal" construction. This idea, introduced in Chapter I for sets and in Chapter II for other concrete categories (such as monoids and lattices), is then developed with successive examples throughout the subsequent chapters.

[1] https://archive.org/details/algebra00macl

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jasomill2y ago

[1] https://archive.org/details/algebra00macl

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