Nicola Carissimi

It is common in category theory to deal with "morphisms between morphisms" (or 2-morphisms), since that is what we have with natural transformations, which are morphisms between functors, e.g.morphisms between morphisms of categories. All this structure can be encoded in the notion of a 2-category, and many concept turn to have a new place where to be studied. Among these, one of the most enjoyable is the concept of adjunction. We will recall these definitions and appreciate how this concept really is "just" 2-categorical.

One problem with the introduction of all this higher level maps is very practical, and related to the way of dealing with (fastly becoming huge and three-dimensional) compositions of 2-morphisms. It becomes then a crucial matter here a clean and intuitive way of representing these creatures, and that is where string diagrams arise (though the history is not that linear and much more interdisciplinary that it may seem!) We will describe their functioning, and time permitting we will see an application.