This
is a first course in Category Theory, with no prerequisites. Examples will
be taken from various areas of mathematics but these will be given as examples
rather than as starting points, so prior knowledge of all examples is not
necessary. Everyone is welcome.
Material I intend to cover
Categories, functors and natural transformations. Examples drawn from different
areas of mathematics.
Equivalence of categories, skeletons.
Representable functors, the Yoneda lemma.
Adjunctions.
Limits and colimits. Preservation and creation.
Density, completion, fibrations.
The Adjoint Functor Theorems.
Kan extensions.
Monads.
The Eilenberg-Moore and Kleisli categories, and their universal properties.
Monadic adjunctions, Beck's Monadicity Theorem.
Enriched categories, internal categories, monoidal categories, bicategories,
n-categories.
For a schedule of what I intend to cover in each class, click
here.
References
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.