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General


This course is part of the MAGIC core.

Description

Category theory is the language of much of modern mathematics. It starts from the observation that the collection of all mathematical structures of a certain kind may itself be viewed as a mathematical object - a category. This is an introductory course in category theory. The main theme will be universal properties in their various manifestations, one of the most important uses of categories in mathematics.
Semester

Spring 2012 (Monday, January 16 to Friday, March 23)

Timetable
  • Mon 10:05 - 10:55

Students


Photo of Bana Al Subaiei
Bana Al Subaiei
(Southampton)
Photo of Abdulsatar Al-Juburie
Abdulsatar Al-Juburie
(Newcastle)
Photo of Kalyan Banerjee
Kalyan Banerjee
(Liverpool)
Photo of Yumi Boote
Yumi Boote
(Manchester)
Photo of Tom Brookfield
Tom Brookfield
(Birmingham)
Photo of Christopher Cave
Christopher Cave
(Southampton)
Photo of Yu-Yen Chien
Yu-Yen Chien
(Southampton)
Photo of Anthony Chiu
Anthony Chiu
(Manchester)
Photo of Michael Dymond
Michael Dymond
(Birmingham)
Photo of Michal Ferov
Michal Ferov
(Southampton)
Photo of Adam Firkin
Adam Firkin
(Birmingham)
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Matthew Gadsden
(Sheffield)
Photo of Belema Gancarz
Belema Gancarz
(Durham)
Photo of Kostas Georgiadis
Kostas Georgiadis
(Loughborough)
Photo of Tom Harris
Tom Harris
(Southampton)
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Andrew Jones
(Sheffield)
Photo of Daniel Jones
Daniel Jones
(Durham)
Photo of Fiachra Knox
Fiachra Knox
(Birmingham)
Photo of Benjamin Lang
Benjamin Lang
(York)
Photo of John Lapinskas
John Lapinskas
(Birmingham)
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Nicholas Loughlin
(Newcastle)
Photo of Umberto  Lupo
Umberto Lupo
(York)
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Keith McCabe
(Durham)
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Gregory McKay
(East Anglia)
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David OSullivan
(Sheffield)
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Laura Phillips
(Manchester)
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Alan READING
(Birmingham)
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Felix Rehren
(Birmingham)
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Andrew Steele
(Nottingham)
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Simon StJohn-Green
(Southampton)
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Dragomir Tsonev
(Loughborough)
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Bryan Williams
(Liverpool)
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Dan Dan Yang
(York)
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Rida-e Zenab
(York)


Prerequisites


There are few formal prerequisites to the material. However, I will be giving examples from mathematics to motivate the ideas and demonstrate how they are used, so an undergraduate degree in mathematics (rather than for example computer science or philosophy) would be an advantage. In particular, I will assume some knowledge of algebra such as vector spaces and their bases, and groups, but undergraduate level knowledge of these subjects is sufficient.

Syllabus


The topics covered are:
  1. Categories: definitions, examples, special kinds of arrows and objects, duality
  2. Functors: definitions, examples, full and faithful functors, subcategories, Hom-functors, contravariant functors
  3. Universal properties: examples including vector space bases, fields of fractions, tensor products, quotients, products, and coproducts
  4. Natural transformations: definitions and examples, functor categories, equivalence of categories, horizontal composition
  5. Limits: examples, general definition, computing limits in Set, complete categories
  6. Colimits: definition, examples, computing colimits in Set
  7. Adjunctions: vector space bases, formal definition, examples, unit and counit
  8. Limit preservation: right adjoints preserve limits
  9. Limit creation: general adjoint functor theorem, examples
  10. The category of Sets

Bibliography


Categories for the working mathematician Mac Lane
Handbook of Categorical Algebra: Basic category theory Borceux
Category Theory Awodey
Note:

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Assessment


The course will be assessed by a single exam. The paper is available under the Assignments tab, and the deadline is 13th April 2012.

Assignments


Exam for MAGIC009: Category Theory Spring 2012

Files:Exam paper
Deadline: Friday 13 April 2012 (737.3 days ago)