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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.

## Subfields

• 18A: General theory of categories and functors
• 18B: Special categories
• 18C: Categories and algebraic theories
• 18D: Categories with structure
• 18E: Abelian categories
• 18F: Categories and geometry

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

• Homological algebra, by Henri Cartan and Samuel Eilenberg. Princeton, Princeton University Press, 1956.
• Homology, Saunders Mac Lane. Berlin; New York : Springer-Verlag, c1995.
• An introduction to homological algebra / Joseph J. Rotman. New York: Academic Press, 1979.
• Hilton-Stammbach
• Weibel, Charles A.: "An introduction to homological algebra", Cambridge University Press, Cambridge, 1994. 450 pp. ISBN 0-521-43500-5 MR95f:18001

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]