This course is a systematic introduction to modern Category Theory, useful to all students in Algebra, Geometry, Topology, Combinatorics, or Logic. Category Theory has come to occupy a central position in contemporary Mathematics and Theoretical Computer Science, and is also applied to Mathematical Physics. Roughly, it is a general mathematical theory of structures. By emphasising “how things relate to each other” rather than “what things are”, it is able to recognise structures across the mathematical sciences and unveil unifying principles such as universal properties. As an illustration (among many), we shall see in this course how the concepts of tensor products, open subsets, sheaves, generating functions and existential quantifiers are all examples of left adjoints, and that general properties of left adjoints are important in all cases.

The general abstract understanding provided by Category Theory is not considered a goal in itself but rather as an advantageous starting point, an important guiding principle, for research into more specific problems. Learning category theory is a shortcut to mathematical maturity. And it is also fun!

This course is adressed to all graduate students in Algebra, Geometry, Topology, Combinatorics, or Logic, as well as bright or particularly motivated undergraduate students in these areas, and abstractly minded students in other areas of mathematics. Postdocs who never had the opportunity to learn category theory systematically, but nevertheless use categorical language, will also benefit from following the course.

In addition to the students in the five core areas mentioned, it is also expected that the course will be of interest by students in certain areas of Theoretical Computer Science and Mathematical Physics, such as Quantum Information Theory and Program Semantics.

The material is organised into two blocks: one of 12 hours, covering basic material (Block B), and one of 8 hours with six additional topics (Block A).

Block B consists of 6×2 hours, closely following [T. Leinster: Basic Category Theory, CUP 2014], one chapter per 2-hour lecture:

B1: Categories, functors and natural transformations.
B2: Adjoints.
B3: Interlude on sets.
B4: Representables.
B5: Limits.
B6: Adjoints, representables and limits.

Block A consists of 8 hours covering the following six additional topics (not covered by Leinster’s book), which serve on one hand to illustrate the basic concepts acquired, and on the other hand to point towards some current developments and applications:

A1: Monoidal categories and graphical calculus Vector spaces, Frobenius algebras, and rudiments of categorical quantum mechanics (after Coecke and Abramsky).

A2: Enriched category theory (2 hours) Abelian categories, metric spaces as enriched categories (after Lawvere); magnitude of metric spaces and other enriched categories, and its geometric and topological significance.

A3: Monads, Lawvere theories and operads (2 hours) Adjunctions and monads, Eilenberg {Moore construction, Kleisli construction, algebraic theories, operads. Examples, including the monads for ultrafilters and probability measures.

A4: Presheaf categories and knowledge representation. Diagram categories and categories of elements; colimits and sheaves; applications to graph theory and database theory (after Spivak).

A5: Species and polynomial functors. Natural numbers and finite sets, power series and species, calculus of species and polynomial functors.

·A6: Groupoids Symmetries as obstruction to classification problems (representability of moduli problems), homotopy solutions in terms of stacks, with a view towards higher category theory and homotopy theory.

(Although the examples in Block A are chosen more on the exotic side (for pedagogical effect), the concepts they illustrate are general, and examples abound also in the five general mathematics subjects areas mentioned, and elsewhere in the mathematical sciences).

For Block B we follow

[T. Leinster: Basic Category Theory, CUP 2014]. Additional references include:[E. Riehl: Category Theory in Context, CUP 2014] [S. Awodey: Category Theory, OUP 2010], and [F. W. Lawvere & R. Rosebrugh: Sets for Mathematics, CUP 2003] (for B3).

For Block A:

A1: [J. Baez & M. Stay: Physics, Topology, Logic and Computation: a Rosetta Stone, 2009] [J. Kock: Frobenius algebras and 2D topological quantum field theories, CUP 2004]. [C. Heunen & J. Vicary: Categorical Quantum Mechanics, CUP, forthcoming,prefinal version available].

A2: [F. Borceux: Handbook of Categorical Algebra: Volume 2, CUP 1994] [F. W. Lawvere: Metric spaces, generalized logic, and closed categories, 1973, TAC Reprint 2002] [T. Leinster: The Magnitude of Metric Spaces, Documenta Math. 2013].

A3: [S. Awodey: Category Theory, OUP 2010] [T. Leinster: Higher operads, higher categories, CUP 2003].

A4: [M. La Palme Reyes, G. Reyes and H. Zolfaghari: Generic figures and their glueings: A constructive approach to functor categories. Polimetrica, 2004] [D. Spivak: Category Theory for the Sciences, MIT Press 2014].

A5: [A. Joyal: Une théorie combinatoire des séries formelles, Adv. Math. 1981] [J. Baez, J. Dolan: From finite sets to Feynman diagrams, Mathematics unlimited, 2001] [J. Kock: Notes on polynomial functors, manuscript, 2009].

A6: [R. Brown: Topology and Groupoids, Booksurge 2006].