Appendix C - Higher monads

Summary

To each pair (p,q) of nonnegative integers, we associate a 2-category whose 0-cells are (p,q)-monads, 1-cells are (p,q)-lax functors, 2-cells are morphisms between (p,q)-lax functors. The starting point is the pair (0,0) which yields the 2-category of all categories whose 0-cells are categories, 1-cells are functors, 2-cells are natural transformations. This is level zero. For the pair (1,0), we have the 2-category whose 0-cells are monads, 1-cells are lax functors, and 2-cells are morphisms between lax functors. For the pair (0,1), there is a similar 2-category involving comonads and colax functors. This is level one. The passage from level zero to level one can be formalized via the monad and comonad constructions. Applying these constructions on level one yields the 2-categories on level two, and so on. Double monads (distributive laws), double lax functors, bimonads (mixed distributive laws), bilax functors, and so on, appear on level two. Higher monad algebras are constructed in a straightforward manner from the 2-categories of higher monads. Here we concentrate on the case of algebras, coalgebras, bialgebras which arise from monads, comonads, bimonads, respectively.