# ESSLLI'01 Course on Coalgebras and Modal Logic

The course notes ( ps.gz, ps) are available in a revised version from October 2001.

Sketch of the 5 Lectures:

• Introduction to Systems. Behavioural equivalence, bisimulation, coinduction, final coalgebras, examples, applications. (No category theory).

• Coalgebra. Introduction to universal coalgebra and, as far as needed, category theory. (In contrast to the notes, we will spend quite some time to introduce the necessary categorical concepts.)

• Introduction to Modal Logic. Just some basics on Kripke models and frames, definability and bismulation.

• Coalgebras and Modal Logic (coalgebras as dynamic systems and modal logics as the appropriate specification languages; an overview of some current approaches).

• On the Duality of Modal and Equational Logic Using a categorical formulation of the coalgebraic semantics of modal logic we can explain the common ground of the different approaches discussed in the previous lecture; moreover, the idea that `modal logic is dual to equational logic' can now be formally justified. As an application we prove the dual of the HSP-theorem (Birkhoff's variety theorem).

Recommended Literature (updated May 2002):

I plan to make the lecture self contained, so it will not be necessary to read any of the following. If you want to prepare a bit nevertheless read on modal logic and on category theory (see below). The remaining references are for the curious ...

• A good introduction to algebras and coalgebras presenting (co)induction, duality, and examples is
B. Jacobs and J.J.M.M. Rutten, A Tutorial on (Co)Algebras and (Co)Induction. Bulletin of EATCS Vol. 62, 1997, pp. 222--259.

Probably the best for a first introduction.

• The standard reference for a theory of universal coalgebra along the lines of universal algebra is
J.J.M.M. Rutten, Universal coalgebra: a theory of systems.Technical Report CS-R9652, CWI, Amsterdam, 1996. A revised version appeared in Theoretical Computer Science 249(1), 2000, pp. 3-80.
This paper contains a thorough study of universal coalgebra and contains background material for further studies.

Another more recent reference for universal coalgebra are the lecture notes
H. Peter Gumm, Elements of the general theory of coalgebras. LUATCS'99, Rand Afrikaans University, Johannesburg, South Africa, 1999.

• In case you are interested in universal algebra an excellent introduction from the computer science point of view is
W. Wechler, Universal Algebra for Computer Scientists, Springer, 1991.

• For background on modal logic I recommend
P. Blackburn, M. de Rijke, Y. Venema, Modal Logic. CUP, 2001.
My ESSLLI-lecture will present what is necessary to know about modal logic in order to understand the relationship to coalgebras. But to really appreciate this relationship (which has only been discovered during the last few years) some additional background on modal logic will be useful: For this read chapters 1.1 - 1.3, 2.1 - 2.3, and note theorem 2.65.

• The discovery of the relation between coalgebra and modal logic can be traced back to the book
J. Barwise, L. Moss, Vicious Circles, CSLI, Stanford 1996.
A very readable and exciting book on paradoxa, non-well founded set theory, coalgebras, and modal logic.

• Concerning category theory, the course will be self-contained. But if you have seen before the definitions of functor, product/coproduct, epi/mono, and preferably also of limit/colimit, natural transformation, and adjunction things will be easier (even if you don't remember the definitions, seeing them again will make you feel more comfortable when they appear in the course). You can consult e.g. one of the following:

Lawvere, Schanuel, Conceptual Mathematics, Cambridge University Press, 1996.
This book teaches to think categorically and avoids technicalities or advanced examples. Take a look!

S. Mac Lane, Categories for the Working Mathematician, Springer 1971.
The classical reference.

J. Adamek, H. Herrlich, G. Strecker, Abstract and Concrete Categories , John Wiley and Sons, 1990.
Presents a lot of material not available in Mac Lane's book.

A. Asperti, G. Longo. Categories, Types and Structures, MIT Press, 1991.
For those who are interested in the relationship between category theory and type theory. Electronically available.

M. Barr, C. Wells, Category Theory for Computing Science, Prentice Hall, 1990.
A very readable introduction with many examples familiar to computer scientists. A shorter introduction is electronically available as the lecture notes of their ESSLLI'99 course.

F. Borceux. Handbook of Categorial Algebra. Cambridge University Press, 1994.
Perhaps not the best starter but most valuable for learning category theory seriously.

B. Pierce, Basic Category Theory for Computer Scientist, MIT Press, 1991.
A small book but still containing all the definitions mentioned above.

R.F.C. Walters, Categories and Computer Science, MIT Press, 1991.
An undergraduate text emphasising objects as data types and arrows as programs.

• Additional list of electronically available lecture notes on category theory:

M. Caccamo, J.M.E. Hyland, G. Winskel: Lecture Notes in Category Theory. BRICS Lecture Series, 2001.

M. Fokkinga: A Gentle Introduction to Category Theory - the calculational approach. University of Utrecht, 1992.

Chris Hillman: A Categorical Primer. August 2001.

Tom Leinster: Category Theory.
This page contains an informal introduction to category theory and, for example, a nice explanation of the Yoneda Lemma.

Jaap van Oosten: Basic Category Theory.

D. Turi: Category Theory Lecture Notes. LFCS, Univeristy of Edinburgh, 2001.

Alexander Kurz