This
is a first course in Category Theory, with no prerequisites. It
is also a course in higherdimensional category theory, leading to the
definitions of higherdimensional 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

Wed
24/10/07

Thurs
25/10/07
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


Thurs
1/11/07


Functors and some examples;
natural
transformations, equivalence of categories

Fri
2/11/07
1.102.00


The
2category Cat,the definition of 2category,
the principle of internalisation, "internal" definition of category
(briefly) 

Wed
7/11/07

Thurs
8/11/07

Monoidal
categories, bicategories 
Monads

Wed
21/11/07

Thurs
22/11/07

Multicategories,
operads, Tmulticategories,
Toperads 
Adjunctions


Mon
26/11/07
4.105.00


Monads
and adjunctions

Wed
28/11/07


Toperads, the free strict omegacategory monad


Wed 5/12/07

Thurs
6/12/07

Batanin's
definition 
Representability
and the Yoneda
Lemma 
Wed
12/12/07

Thurs 13/12/07

The
nerve
construction, Segal categories 
Limits
via representability,
Cartesian closed categories 
Wed 19/12/07

Thurs
20/12/07

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,
SpringerVerlag,
second edition 1998. Still the best onevolume book on the subject,
written
by one of its founders.
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
ncategories,
Advances in Mathematics 136, no. 1, 39103 (1998), also
available
here
(I can't link to Advances at the
moment because their site is down)
and Tamsamani's appeared in:
For a concise expository account of these and other definitions see:
For a chatty and unconcise expository account see:
Cheng, E., and Lauda, A.,
Higherdimensional categories: an
illustrated guidebook, available
here.
For more on generalised operads, the free strict omegacategory monad,
and various other higherdimensional 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