Part III Category Theory

Michaelmas/Autumn/Fall 2000, 24 lectures


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:

  • 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.
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.


  • 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 background.
  • 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.

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