Category theory

This department is a part of the Subdivision of Higher Algebra



Department description
Category theory is a relatively new birth that arose from the study of cohomology in topology and quickly broke free of its shackles to that area and became a powerful tool that currently challenges set theory as a foundation of mathematics, although category theory requires more mathematical experience to appreciate and cannot in its current state be reasonably used to introduce mathematics. It has also found many applications in the physical sciences. The goal of this department is to familiarize the student with the theorems and goals of modern category theory. Prerequisites to full appreciation of this area of study includes knowledge of topology up to basic homology theory, and some basic idea of cohomology.

Department news

 * Monday, August 28, 2006 - Department founded!

Learning materials and learning projects
Introduction to Category Theory

Offsite Learning Materials

 * Saunders Mac Lane, the Knight of Mathematics
 * Bibliography for Category Theory
 * Ronald Brown. 2010. "Nonabelian Algebraic Topology: filtered spaces, crossed complexes and cubical homotopy groupoids." textbook
 * G.M. Kelly, F.A.A. 2005."Basic Concepts of Enriched Category Theory"
 * P. J. Higgins. 2005 . "Categories and Groupoids.", TAC Reprint pp.195.
 * Ibid.- Secondary source
 * Andrea Asperti and Giuseppe Longo. 1991."Categories, Types and Structures: Category Theory for the working computer scientist", pp.301.

Recommended Reading Material
If you want to learn the way mathematicians do, you will need to read and complete the exercises in at least one of the following books. While this site provides a supportive community of peers and teachers, nothing beats having a well-organized and well-written text that you can study anywhere to learn from the masters. Mathematics is not a spectator-sport. You must do mathematics to learn it.

For those with less background:
 * Robert Goldblatt. (2006). Topoi: The Categorial Analysis of Logic (Revised Edition). Dover Publications. ISBN 0486450260. This book is simultaneously gently paced, broad and often deep. It introduces category theory, contrasts it with set theory and then develops logic in the language of category theory. However, unlike most books that deal with such matters, the book introduces the concepts informally. Any formal treatment is reserved for later in the book or relegated to more advanced textbooks. One essentially gets a broad understanding of some of the deepest areas in mathematics at a heavily discounted price.


 * Barr & Wells. (1999). Categories for Computing Science (third edition). Centre de Recherches Mathématiques. ISBN 2921120313.  Info including ordering here.  A mostly gentle pace, 100s of solved exercises and reasonable price tag make this one a must-have for people without a formal teacher.


 * Lawvere & Schanuel. (1997). Conceptual Mathematics: A First Introduction to Categories Cambridge University Press. ISBN 0521010608


 * B.C. Pierce (1991) Basic Category Theory for Computer Scientists MIT Press. ISBN 0262660717. (Short and concise introduction.  Probably not enough exercises for most people to really learn it from.)

And requiring somewhat more:
 * Lawvere & Rosebrugh. (2003). Sets for Mathematics Cambridge University Press. ISBN 0521478170 (set theory axiomatized through category theory, for advanced undergraduates and beginning graduates; more advanced and then Lawvere & Schanuel.  Starts fairly easy but seems to get harder.)


 * Awodey, Steven (2006) Category Theory (Oxford Logic Guides 49) Oxford University Press. ISBN 0198568614 (very elegant, from what AA has seen of the no-longer-available downloadable pdf version, but perhaps a bit demanding for teacherless beginners with modest talent).


 * Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Springer. ISBN 0-387-98403-8. (on the whole, not for beginners to read consecutively, but some of the discussions, such as the one of adjunctions, are very useful even for novices. '10% comprehension of Mac Lane is worth 90% of almost anybody else', I read somewhere ).

Wikipedia

 * Category theory

Active participants
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this department, you can list your name here (this can help small departments grow and the participants communicate better; for large departments a list of active participants is not needed).


 * User:Bci21
 * User:X