Category Theory

PMA 6308

Autumn 2007
Wednesday 1.10-2.00
Thursday 3.10-4.00
J11 unless specified

Eugenia Cheng
Office: J24

This is a first course in Category Theory, with no prerequisites.  It is also a course in higher-dimensional category theory, leading to the definitions of higher-dimensional category by Batanin (and variants) and Tamsamani (and variants). 

In general Wednesday will be a type B lecture and Thursday will be a type A lecture.  Those who already know some category theory should be able to attend only type B lectures and get a coherent exposition....

Note that some dates will change, as will some rooms.  Also some extra type A lectures may be added to make up the 20 hours for RTP credit.

Printed notes
Some old notes are linked at the bottom of the page; I am aiming to make new notes which are rather less terse and more chatty and explanatory.  I will make the drafts available here as I go along.  They may or may not correspond to lectures... There will be exercises at the end of each section, with answers provided eventually.  If you particularly want answers let me know because that will motivate me to provide them!

Click on the links for pdf files.  Please remember that these are drafts!  In particular, please report errors/typos. Thanks.

Section 1:  Categories, definitions and examples (roughly lecture 1)
Section 2:  Some basic universal properties (roughly lecture 2)
Section 3: Functors (the first half of lecture 3, expanded)

Schedule of topics

This is the schedule of what I'm aiming to cover when.  This is likely to change as it goes along.  Actual material covered will be in straight text after the event; intended material will be in italics until the class has occurred.

Type B
Type A
in LT11
Introduction, overview, definition of category and examples, definition of functor (not really B)
Universal properties, some basic limits, examples in Set: initial and terminal objects, (co)products, pullbacks and pushouts, (co)equalisers


Functors and some examples; natural transformations, equivalence of categories

The 2-category Cat,the definition of 2-category, the principle of internalisation, "internal" definition of category (briefly)
Monoidal categories, bicategories Monads
Multicategories, operads, T-multicategories, T-operads Adjunctions


Monads and adjunctions

T-operads, the free strict omega-category monad

Wed 5/12/07
Batanin's definition Representability and the Yoneda Lemma
Thurs 13/12/07
The nerve construction, Segal categories Limits via representability, Cartesian closed categories
Wed 19/12/07
Tamsamani's definition Preservation of limits, adjoint functor theorems
extra lectures?

Model categories

References for Part A

1. F. Borceux, Handbook of Categorical Algebra, Cambridge U.P., 1994. Three volumes which together provide perhaps the best modern account of everything you should know about category theory: volume 1 covers most but not all of this course.

2. S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, second edition 1998. Still the best one-volume book on the subject, written by one of its founders.

Online notes

This course will be a variant of ones I have taught before in Cambridge and Chicago.  Notes for the Cambridge course were typed up by Richard Garner.  Please note that these were originally just for his own personal use.  Also, the current course will be rather different, but almost everything in part A is covered in these notes.

notes in pdf

Video notes

For video lectures in 10 minute chunks, have a look at The Catsters (aka Eugenia Cheng and Simon Willerton) on YouTube, here.

References for Part B

Batanin's definition originally appeared in:

Batanin, M., Monoidal globular categories as a natural environment for the theory of weak n-categories, Advances in Mathematics 136, no. 1, 39--103 (1998), also available here (I can't link to Advances at the moment because their site is down)

and Tamsamani's appeared in:

Tamsamani, Z., Sur des notions de n-categorie et n-groupoide non strictes via des ensembles multi-simpliciaux, K-Theory 16 (1999), no. 1, 51--99, also e-print alg-geom/9512006.

For a concise expository account of these and other definitions see:

Leinster, T., A survey of definitions of n-category, Theory and Applications of Categories 10 (2002), no. 1, 1-70, also e-print math.CT/0107188.

For a chatty and unconcise expository account see:

Cheng, E., and Lauda, A., Higher-dimensional categories: an illustrated guidebook, available here.

For more on generalised operads, the free strict omega-category monad, and various other higher-dimensional topics, see:

Leinster, T., Higher Operads, Higher Categories, LMS Lecture Note Series 298, CUP, available at math.CT/0305049.

Page last modified 4th December 2007, 12:38