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18: Category theory, homological algebra


Introduction

Category theory, a comparatively new field of mathematics, provides a universal framework for discussing fields of algebra and geometry. While the general theory and certain types of categories have attracted considerable interest, the area of homological algebra has proved most fruitful in areas of ring theory, group theory, and algebraic topology.

History

A survey article which discusses the roles of categories and topoi in twentieth-century mathematics.

Applications and related fields

The word "category" is used to mean something completely different in general topology. [Schematic of subareas and related areas]

Subfields

This is among the smaller areas in the Math Reviews database.

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

A full, wide-ranging text on category theory is by Borceux, Francis: "Handbook of categorical algebra", 3 vol (1: Basic category theory; 2: Categories and structures; 3: Categories of sheaves) (Encyclopedia of Mathematics and its Applications, 50-2.) Cambridge University Press, Cambridge, 1994. 345+443+522 pp. ISBN 0-521-44178-1, 0-521-44179-X, 0-521-44180-3 MR96g:18001

Much more informal is MacLane, Saunders: "Categories for the working mathematician", Springer-Verlag, New York-Berlin, 1971

There are a number of textbooks on homological algebra which should be accessible to graduate students in algebra and topology, such as

There is an excellent, if somewhat dated, collection of "Reviews in Topology" by Norman Steenrod, a sorted collection of the relevant reviews from Math Reviews (1940-1967). Many now-classical results date from that period. Several of the sections cover homological algebra well.

Notes for a course in homological algebra [Lee Lady]

Software and tables

Other web sites with this focus

Selected topics at this site


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Last modified 2004/01/04 by Dave Rusin. Mail: