Martti Karvonen: "Dagger categories: monads and limits"
Talk given as part of Categories Logic and Physics Scotland: http://conferences.inf.ed.ac.uk/claps....
A dagger category is a category equipped with a dagger: a contravariant involutive identity-on-objects endofunctor. Such categories are used to model quantum computing and reversible computing, amongst others. The philosophy when working with dagger categories is that all structure in sight should cooperate with the dagger. This causes dagger category theory to differ in many ways from ordinary category theory. Standard theorems have dagger analogues once one figures out what "cooperation with the dagger" means for each concept, but often this is not just an application of formal 2-categorical machinery or a passage to (co)free dagger categories. We will discuss two instances.
First, as soon as a monad on a dagger category satisfies the Frobenius law, everything works as it should. Dagger adjunctions give riseto such monads. Conversely, such monads factor as dagger adjunctions in two canonical ways; however, the Eilenberg-Moore category needs to be adapted t o inherit the dagger.
Second, limits in dagger categories should be unique up to an unique unitary, that is, an isomorphism whose inverse is its dagger. We rework an initial attempt to a more elegant and general theory. It works well when the diagram has a dagger; however, the formulation of limits using adjunctions needs to be adapted. We will discuss work on defining dagger limits of general diagrams.
Finally, we will discuss formally how dagger categories, while perhaps slightly evil, are not all that bad.