I am a training Algebraic Geometer, and am currently trying to prepare a course on category theory. I would be really thankful if people could tell me about "real-math" applications of theorems about monads which they came across during their career.



Why are monads useful?
The abstract algebra of monads is similar to the algebra of monoids, so that constructions on monoids often suggest similar constructions on monads. This applies in particular to bar constructions. 


A pretty striking application in its time was the use of bar constructions on monads to construct deloopings. Around 1963, Stasheff identified the extra structures on H-spaces $X$ that need to hold in order to construct a single delooping $X \simeq \Omega Y$ compatible with the H-space structures. This was at a time before there were such things as operads and their associated monads. But by the time May's The Geometry of Iterated Loop Spaces had appeared, there was appreciation of the algebra of operads and monads to economically package the panoply of algebraic operations that obtain on iterated loop spaces. In particular, the two-sided bar constructions for monads and their algebras, similar to the Milgram bar construction for monoids, were used to give conceptually simple constructions for iterated deloopings, extending the constructions of Stasheff to much more general contexts. 


