Elements of Category Theory 1000-1M07ET
Most of the lecture will be an introduction to Category Theory, covering the following notions and theorems: categories, functors, natural transformations, equivalence of categories, representable functors, The Yoneda Lemma, limits, colimits, adjoint functors, GAFT, SAFT, cartesian closed categories, presheaf categories, monads, Eilenberg-Moore and Kleisli algebras, Beck's Theorem. The lecture will be illustrated by examples taken mostly from algebra, topology and logic.
In the remaining part of the lecture I intend to discuss Grothendieck toposes from various points of view: as generalized topological spaces, universes of 'sets', and geometric theories. The participants' interests may substantially influence the choice of the material covered in this part.
The course will end with a written exam.
Course coordinators
Type of course
Assessment criteria
The grading will be made on the basis of
1. Active participation in class
2. Written solutions of a set of problems
3. Oral exam
Bibliography
General introduction:
S. MacLane, Categories for the Working Mathematician,
M. Barr, Ch. Wells, Category Theory for Computing Science
Topos Theory:
I. Moerdijk, S. MacLane, Sheaves in Geometry and Logic
M. Barr, Ch. Wells, Toposes, Triples and Theories
Handbooks:
P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium
F. Borceux, Handbook of Categorical Algebra