nLab monad transformation

Idea

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

author(s)	1-morphisms	2-morphisms
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” ${}^1$	—
Lack & Street 2002	“monad morphisms”	“monad transformations”
—	“monad 1-morphisms”	“monad 2-morphisms”
—	“lax transformations” of monads	“modifications” of monads

${}^1$: 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 $Cat$) monad morphisms are, in any case, natural transformation of functors, respecting their monad structure.

Similarly, if one thinks of monads as lax functors $\ast \to Cat$ (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).

Related entries

• monad transformer

• transformation of adjoints

References

• Jean-Marie Maranda, On Fundamental Constructions and Adjoint Functors, Canadian Mathematical Bulletin 9 5 (1966) 581-591 [doi:10.4153/CMB-1966-072-9]

• Jean-Marie Maranda, Sur les Proprietes Universelles des Foncteurs Adjoints, In: Études sur les Groupes abéliens / Studies on Abelian Groups Springer (1968) [doi:10.1007/978-3-642-46146-0_16]

• Armin Frei, Some remarks on triples, Mathematische Zeitschrift 109 (1969) 269–272 [doi:10.1007/BF01110118]

• Dieter Pumplün, Eine Bemerkung über Monaden und adjungierte Funktoren, Mathematische Annalen 185 (1970) 329-337 [eudml:161964, pdf]

• Ross Street, The formal theory of monads, Journal of Pure and Applied Algebra 2 2 (1972) 149-168 [doi:10.1016/0022-4049(72)90019-9]

• Eugenio Moggi, An abstract View of Programming Languages, LFCS report ECS-LFCS-90-113 (1989) [web, pdf]

• Francis Borceux, Def. 4.5.8 in: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]

• David A. Espinosa, Building Interpreters by Transforming Stratified Monads (1994) [pdf, pdf]

• David A. Espinosa, Semantic Lego, PhD thesis, Columbia University (1995) [pdf, pdf, slides:pdf, pdf]

• Sheng Liang, Paul Hudak, Mark Jones, Monad transformers and modular interpreters, POPL ‘95 (1995) 333–343 [doi:10.1145/199448.199528]

• Stephen Lack, Ross Street, The formal theory of monads II, Journal of Pure and Applied Algebra 175 1–3 (2002) 243-265 [doi:10.1016/S0022-4049(02)00137-8]

• Sergei Winitzki, Section 14 of: The Science of Functional Programming – A tutorial, with examples in Scala (2022) [leanpub:sofp, github:sofp]

category: disambiguation