Masterstudium FS 2008: "Introduction to Category Theory"

Lecture: Introduction to Category Theory

Schedule: Tuesdays 14-16 (this can be changed under pression of participants)

Place: Rheinsprung 21, Mathematisches Institut Basel, Kleiner Hörsaal

First lecture: Tuesday, 19. February 2008

For: Mathematicians, physicists, computer scientists, logicians, biologists, neuroscientists, philosophers, etc.

Prerequisites: Grundstudium 1st-2nd year, or nothing

Credit points: 4 (how to get them?)

Language: English

Module: Masterstudium, Weitere Lehrveranstaltungen

Important: If you are attending the seminar please send me an email!

Description:

The aim of this lecture is to explain the basic concepts of category theory to a quite large audience. This theory, initiated by the work of Eilenberg and Mac Lane in 1942, started as a convenient language, well suited for group theory and algebraic topology problems. It then evolved into a theory under the impulse of Grothendieck, Kan, Quillen during the 50's and has been since then successfully applied in many mathematical areas. Since the 60's it entered the realm of the foundations of mathematics and logic thanks to Lawvere and Tierney among others. Nowadays it is touching theoretical physics, computer sciences, cognitive sciences, linguistics, artificial intelligence, etc.

In the lecture we will introduce the fundamental ideas of categories, functors, natural transformations, universal constructions, adjoint functors, groupoids, etc. We will also give many examples arising from mathematics and which are useful for "the working mathematician", as Saunders Mac Lane would say...On demand, we could touch topics like logical aspects of category theory, or even higher dimensional category theory in order to be able to understand the ideas of John Baez and others in mathematical physics.

The exercices are replaced by a small informal seminar. In this seminar, according to the audience, we could embrace examples coming from the above mentioned applications.

"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 and of systems of structures. As category theory is still evolving, its functions are correspondingly developing, expanding and multiplying. At minimum, it is a powerful language, or conceptual framework, allowing us to see the universal components of a family of structures of a given kind, and how structures of different kinds are interrelated. Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth. It can be applied to the study of logical systems in which case category theory is called "categorical doctrines" at the syntactic, proof-theoretic, and semantic levels. Category theory is an alternative to set theory as a foundation for mathematics. As such, it raises many issues about mathematical ontology and epistemology. Category theory thus affords philosophers and logicians much to use and reflect upon. [...]"

Quoted from the Stanford Encyclopedia of Philosophy

Literature:

Mac Lane, S.: Categories for the working mathematician, Springer-Verlag, GTM 5, 1971, second edition: 1998.

Goldblatt, R.: Topoi, the categorical analysis of logic, North-Holland, 1984 (available online).

Bucur, I. & Deleanu, A.: Introduction to the theory of categories and functors, Pure and Applied Maths, Vol. XIX, 1968.

Lawvere, F. W. & Schanuel, S.: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, 1997.

For a more complete (and annotated) bibliography and some historical remarks see the above cited Stanford entry.

Some other links:

The wikipedia entry page for category theory
The Category Theory Research Center
John Baez's Stuff

Credit points:

Students desireful of credit points should attend 80% of the lecture and of the seminar. In addition they should contribute at least once in the below described seminar.

Exercices/Seminar:

The exercice hours will be replaced by a small seminar on categorical notions and applications of category theory to various areas. Participants are encouraged to pick one subject of their interest and to give an account on it. This could range from computer science to linguistics, including of course standard applications to mathematical notions.

Schedule: Tuesdays and/or Thursdays 16-18
Place: Mathematisches Institut Basel
First seminar: Tuesday 26 February (organisation of the seminar)
Assistants: None, for questions please contact me!
Language: Any language is tolerated

If you are interested in attending this lecture or the seminar please contact me by email.

Here is a small sample of possible seminar subjects:

Choose your favorite category, explain some categorical constructions in it and give some functors in/out of it.

Expose a topic from Mac Lane's book (e.g. adjoint functors, abelian categories, ...) or any other related book.

Pick one week in "This Week's Finds in Mathematical Physics" and explain some ideas.

Explain some concepts contained in the following articles (more on demand):

Baez, J. & Dolan, J., "Categorification", 1998.

Baez, J. & Dolan, J., "From finite sets to Feynman diagrams", 2000.

Baez, J., "An introduction to n-Categories", 1997.

Brown, R., "From groups to groupoids", 1986.

Brown, R., "Out of line", 2006.

Grothendieck, A., "Sur Quelques Points d'algèbre homologique", Tohoku Mathematics Journal, 9, 119-221, 1957.

Kan, D. M., "Adjoint Functors", Transactions of the American Mathematical Society, 87, 294-329, 1958.

Brown, R. & Porter, T., "Category theory and higher dimensional algebra: potential descriptive tools in neuroscience", 2003.

Arzi-Gonczaworski, Z., 1999, "Perceive This as That - Analogies, Artificial Perception, and Category Theory", Annals of Mathematics and Artificial Intelligence, 26, no. 1, 215-252.

Baianu, I. C., 1987, "Computer Models and Automata Theory in Biology and Medecine", in Witten, Matthew, Eds. Mathematical Modelling, Vol. 7, 1986, chapter 11, Pergamon Press, Ltd., 1513-1577.

Ehresmann, A. C. & Vanbremeersch, J-P., 1987, "Hierarchical Evolutive Systems: a Mathematical Model for Complex Systems", Bulletin of Mathematical Biology, 49, no. 1, 13-50.

Healy, M. J., 2000, "Category Theory Applied to Neural Modeling and Graphical Representations", Proceedings of the IEEE-INNS-ENNS International Joint Conference on Neural Networks: IJCNN200, Como, vol. 3, M. Gori, S-I. Amari, C. L. Giles, V. Piuri, eds., IEEE Computer Science Press, 35-40.

"In the years between 1920 and 1940 there occurred, as you know, a complete reformation of the classification of different branches of mathematics, necessitated by a new conception of the essence of mathematical thinking itself, which originated from the works of Cantor and Hilbert. From the latter there sprang the systematic axiomatization of mathematical science in entirety and the fundamental concept of mathematical structure. What you may perhaps be unaware of is that mathematics is about to go through a second revolution at this very moment. This is the one which is in a way completing the work of the first revolution, namely, which is releasing mathematics from the far too narrow conditions by 'set'; it is the theory of categories and functors, for which estimation of its range or perception of its consequences is still too early..."

Back to my homepage