The past 15 years have seen an explosion in the use of Category Theory in mainstream Algebraic Geometry. The ideas go back to prescient work of Grothendieck and his co-workers in the 1960's and 1970's but they are now being used to help us to more fully understand the spaces and structures which arise naturally in problems in Algebraic Geometry. My aim will be to try to motivate the development of the categorical tools we now use and to give an insight into one of the current areas of research using notions of triangulated categories and t-structures.

We introduce a generalisation of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed lambda calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between monads and relative monads. Arrows are also an instance of relative monads.

This is joint work with James Chapman and Tarmo Uustalu, recently presented at FoSSaCS 2010.

The talk will look at some examples of how you can learn and gain confidence in many areas of maths by paying attention to the adjunctions that exist or don't exist. I learnt a lot of what I know about group cohomology this way.

We describe a uniform approach to the proof theory of a large class of modal logics, that subsumes intuitionistic modal logic, modal logics over fuzzy sets and probabilistic modal logics that are interpreted over measurable spaces, in an abstract categorical setting. The main contribution is a generic notion of proof system that encompasses e.g. sequent calculi and equational logic. In particular, the sequent calculus formulation leads to generic decidability results.