Aspects of commutative 2-algebra

Description: The aim of the project is to develop further the theory of 2-categories and bicategories, focusing on the analogues of standard notions of commutative algebra (monoid, group, ring, module). Just as commutative algebra provides a useful basis for the development of algebraic geometry, its 2-categorical analogue suggests the development of a 2-algebraic geometry. Both lines of research will be explored.

Some work in this line of research has been done by the PhD supervisor in collaboration with Fiore, Hyland, Winskel and with Joyal, as well as by others, but much remains to be explored. In particular, one would like to extend the classical theory of commutative monads of Kock to relative pseudomonads, building on existing work on commutative 2-monads, by Hyland and Power and by Bourke. Applications of this research can be explored either in mathematical logic (via models of type theories) or algebra (via the programme of categorification).