In 1998, with the support of the European Union and the Government of Canada, mathematicians from four European and three Canadian universities embarked on a four-year program promoting, on the one hand, transatlantic student exchanges and, on the other hand, a collaborative research project on the coordinated teaching of various topics in pure mathematics, the common thread of which is the active use of categorical foundations and methods. Primarily during group meetings held in the summers of 1999 in Coimbra (Portugal), 2000 in Toronto (Canada), and 2001 in Barisciano (Italy), small teams were formed to work on a variety of themes of current interest for which an adequate modern treatment in the literature was not available. These teams included not only faculty members of the sponsoring universities, but also graduate students of these universities who were participating in the exchange program, as well as visiting researchers to the partner universities.

The result of this collaboration is documented in this book which provides a categorical introduction to some key areas of modern mathematics, readable by anyone with a modest knowledge of category theory. While the degree of required general mathematical maturity and particular categorical expertise may vary among chapters, graduate students and researchers should have no difficulty following the main thrust of the theories presented, which in many instances should stimulate further research.