MATH 6180

01/04/17

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Category Theory

Basic Course Information

I'll teach this course in the WinterTerm of 2016-17. Classes run MWF 11:30-12:30 in Vari Hall 2000, starting January 6, 2017.

Mini-Calendar Description:

Originally designed to capture interactions between algebra and topology, the language and theory of categories, functors and natural transformations nowadays permeates significant parts of modern pure mathematics, as well as of theoretical computer science and physics. On one hand, category theory can provide overarching results applicable in multiple fields. Typically, objects in a large category of interest are being investigated through their interaction with their peers, or their functorial footprint in other categories. On the other hand, category theory can contribute to the study of a specific mathematical object, by associating with it a small category, which may then be subjected to a categorical investigation. The course will cover basic categorical concepts and constructions, like limits and colimits, adjoint functors, Yoneda embedding and Kan extensions, and show their utility in familiar categories (like those of ordered sets, groups, or topological spaces). Time permitting, a glance at more special topics, like monad and topos theory, will be provided toward the end of the course. While there is no prerequisite course, students are expected to have a solid undergraduate pure math background, in particular in algebra, and preferably also in topology.

If you are not sure whether this course is suitable for you, please contract me, by email, or make an appointment.

   

     Lecture Topics

n   Categories, functors, natural transformations

n   Adjoint functors, examples from set theory, algebra and topology

n   Limits and colimits, completeness and cocompleteness

n   Yoneda Lemma, Yoneda embedding, Kan extensions

n   Cartesian closedness, toposes

n   Factorization systems, fibrations, topological functors

 

Individual Project Topics

n   Monads, monadicity criteria

n   Monoidal categories, enriched category theory

n   Abelian and semi-abelian categories

n   Monad-quantale-enriched categories

 

There is no single recommended text for the course, but see below for a list of recommended books.

Evaluation will be based on

- 5 homework assignments, to be counted at 10% each toward your final grade,

- an individual project to be counted at 20%,

- a final test, counting 30%.

 

Recommended Books:

Tom Leinster, Basic Category Theory, Cambridge University Press 2014, available on line at https://arxiv.org/abs/1612.09375

Steve Awodey, Category Theory, Oxford University Press 2006.

Francis Borceux, Handbook of Categorical Algebra 1-3, Cambridge University Press 1994.

Saunders Mac Lane, Categories For the Working Mathematician (2nd Edition), Springer 1997.

F. William Lawvere and Stephen H. Schanuel, Conceptual Mathematics, A first introduction to categories, Cambridge University Press 1997.

Jiri Adamek, Horst Herrlich and George E. Strecker, Abstract and Concrete Categories, The Joy of Cats, Wiley 1990; available on-line at http://tac.mta.ca/tac/reprints/articles/17/tr17abs.html .

Michael Barr and Charles Wells, Toposes, Triples and Theories, Springer 1985; available on line at http://tac.mta.ca/tac/reprints/articles/12/tr12abs.html .

Dirk Hofmann, Gavin J. Seal, Walter Tholen (editors), Monoidal Topology, A Categorical Approach to Order, Metric and Topology, Cambridge University Press 2014. Preliminary version

 

The University has strict policies regarding academic dishonesty, please see:

http://www.yorku.ca/secretariat/policies/document.php?document=69

 

 

Unofficial Grades (final digits of your student number)

Student # Grade Student # Grade Student # Grade
           
           
           
           
           
           
           
           
           

 

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