Since the time of Noether,
mathematicians have learned the lesson that when faced with a class of
structures, it is almost always fruitful to consider the
structure-preserving maps: in other words, to work with a category
rather than a mere set or class. More recently, it has been realized
that it is often useful to go further: don't just take objects and maps
between them, but also maps between maps, maps between maps between
maps, and so on. This is not obvious (what is a map between maps,
anyway?) but turns out to be an idea of incredible fertility. I will
concentrate on what happens when you go just one step further, from
ordinary categories to 2-categories or bicategories. I will also
explain how the bicategorical approach sheds light on the notion of
Morita equivalence.