I would like to end this lecture by just giving a very brief glimpse of how all this can be understood and explained using a very abstract theory which is category theory. Category theory can be seen as the generalisation of set theory and partial order theory. As we see a purely set theoretical approach is viable as the one we carried out earlier, but very often we have to deal with inessential implementation issues. Now, for example in defining certain streams and functions over streams we really have to get down to the fine details. In the categorical approach we can abstract from many of these details, and moreover, the categorical approach allows us to be very suggestive and to correlate induction and co-induction in terms of duality. So here I only have time to illustrate the idea again using some conceptual ratios. So the relationship running between monotone operator and functor is the same that runs between a pre-fixed point and an algebra, but least fixed point and an initial algebra. Monotone operator or a functor are post-fixed point and a co-algebra and a maximal fixed point and a final co-algebra. We can view co-induction as an example of co-recursion. Yes, let me just illustrate this co-algebraic approach by pointing out the following conceptual ratio. The relationship running between monotone operator and functor is the same relationship that runs between pre-fixed point and algebra. The same relationship that we have between least fixed point and initial algebra. On the dual side we will have that this relationship is precisely the relationship between post fixed point and co-algebra and maximal fixed point and final co-algebra. Moreover, category theory allows for a treatment of co-induction vis-à-vis co-recursion, which is less ad hoc with respect to the purely set theoretical approach. This perspective has initiated a branch of semantics of concurrency which is called final semantics. Of course this brings us to an end of this second lecture and of course I would just like to just mention that the fact that on the newspaper there was the solution to the exercise that I had set in class of course it was a very peculiar coincidence, but still I think we should consider it as a very positive omen to the course.

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