Category theory begins with the observation that the collection of all
mathematical structures of a given type, together with all the maps
between them, is itself an instance of a nontrivial structure which can
be studied in its own right. In keeping with this idea, the real objects
of study are not so much categories themselves as the maps between them -
functors, natural transformations and (perhaps most important of all)
adjunctions. Category theory has had great success in the unification
of ideas from different areas of mathematics; it has now become an
indispensable tool for anyone doing research in topology, abstract algebra,
mathematical logic or theoretical computer science (to name but a few).
This course aims to give a general introduction to the language of category
theory, and should therefore be of interest to a large proportion of pure
Part III students.
Lectures will cover:
Prerequisites are very few. Many of the examples
will be drawn from algebra and topology, so a very basic acquaintance with
these areas would be helpful; but background can be supplied as we go along.
A. Categories, functors and natural transformations. Examples
drawn from different areas of mathematics. Faithful and full functors,
equivalence of categories.
B. Adjoints. Examples. Description in terms of the triangular
identities, and in terms of comma categories. Uniqueness of adjoints.
C. Representables. Representation of functors. The Yoneda
embedding and the Yoneda lemma.
D. Limits and colimits. Examples. Construction of limits
from products and equalizers. Cartesian closed categories.
Preservation, reflection, and creation of limits. Categories of elements,
and every presheaf as a colimit of representables.
E. The Adjoint Functor Theorems. Applications.
F. Monads. The monad induced by an adjunction. The Eilenberg-Moore
and Kleisli categories, and their universal properties. Monadic adjunctions
and the Monadicity Theorem. Examples.
Saunders Mac Lane, Categories for the Working Mathematician, Springer
(1971). Still the best all-round book on the subject, written by
one of its founders.
Francis Borceux, Handbook of Categorical Algebra, Cambridge University
Press (1994). Three volumes which together provide a comprehensive
modern account of category theory. Clearly written, but the emphasis often
differs from that of this course.
Colin McLarty, Elementary Categories, Elementary Toposes, Oxford
University Press (1992), chapters 1-12 only. A gently-paced introduction
to categorical ideas, written by a philosopher for those with little mathematical
F. William Lawvere, Stephen H. Schanuel, Conceptual Mathematics,
Cambridge University Press (1997). An intriguing experiment: category theory
for high school students, complete with classroom dialogues. Not a textbook
for the course, but worth a flick through.
You can go back to the Part III Category Theory