Natural transformations, functors and categories, the fundamental notions of
Category theory (CT), were introduced by Eilenberg and Mac Lane in their
development of algebraic topology and homological algebra. The categorical
concepts of sheaves and Kan extensions soon became key notions of algebraic
topology. Kan also introduced adjoint functors, and Lawvere discovered
that logical rules of inference are instances of them. Later Grothendieck
and his school developed the notion of topos in algebraic geometry, which,
surprisingly, became a categorical framework of higher-order logic and set
theory.
Besides the extensive use of CT in mathematics, CT has applications
to many areas of theoretical computer science such as software engineering, ar-
tificial intelligence and automata theory. As CT is a theory of a general
notion of “function”, it soon became appropriate for the semantic investiga-
tion of programs. Categories provide semantics for type theories and type
theories can be seen as the formal syntax for categories.
The following topics are among the ones to be studied in the seminar: cat-
egories, functors and natural transformations, limits and colimits, cartesian
closed categories, the Yoneda lemma, the Grothendieck construction, the
density of representable functors, adjoint functors, monads, the Chu con-
struction, type theories and categories.
This seminar is addressed to computer science master students with a background in mathematics, as covered for instance in the “Informatik plus Mathematik” Bachelor degree or a similar programme with some component of abstract mathematics.
Each participant will give two 30 minutes talks, one in the first half of the seminar and one in the second.
Additionally, every participant is expected to submit a written piece of work after the end of the teaching term. The grading will be based on both the talk and the write-up.
The following sources can be used:
J. Lambek, P. J. Scott: Introduction to higher order categorical logic, Cam-
bridge University Press, 1986.
S. Awodey: Category Theory, Oxford University Press, 2010.
M. Barr, C. Wells: Category Theory for Computing Science, Reprints in
Theory and Applications of Categories, No. 22, 2012.
E. Riehl: Category Theory in Context, Aurora: Dover Modern Math Orig-
inals, 2016.
Place and time:
Wednesday 14-16, Central Building, DZ007, Start: 20.10.2021
The schedule and the content of presentations are expected to be determined in the first meeting