nLab monad transformation




The following terminology for the (higher) morphisms in the 2-category of monads have appeared in the literature or are otherwise sensible:

Maranda 1966, 1968,
Frei 1969,
Pumplün 1970,
Borceux 1994
“morphisms of monads”
Street 1972“monad functors”“monad functor transformations”
Moggi 1989“monad-morphisms”
Espinosa 1995,
Liang, Hudak & Jones 1995
monad transformers 1{}^1
Lack & Street 2002“monad morphisms”“monad transformations”
“monad 1-morphisms“monad 2-morphisms
lax transformations” of monadsmodifications” of monads

1{}^1: Notice here that the “monad transformers” in Espinosa 1995, Liang, Hudak & Jones 1995 (as commonly understood now in Haskell) are indeed 1-morphisms of monads, but understood with additional structure, namely appearing in natural families constituting a pointed endofunctor on the category of monads (made explicit in Winitzki 2022 p. 474).

Of course, monads are often referred to via their underlying functors, the morphisms between which are, of course, commonly known as transformations, too: natural transformations. Therefore (for the usual monads in Cat Cat ) monad morphisms are, in any case, natural transformation of functors, respecting their monad structure.

Similarly, if one thinks of monads as lax functors *Cat\ast \to Cat (which is where they get their name from!, see here) then their 1-morphisms are by default to be called lax natural transformations (and their 2-morphisms modifications).


category: disambiguation

Last revised on September 22, 2023 at 06:43:54. See the history of this page for a list of all contributions to it.