The classical ``monadicity theorem'' categorifies the idea that algebraic structures are often characterized as given by quotients of free objects of the requisite sort. One starts with an adjoint pair of functors and asks whether the target of the left adjoint is equivalent to the category of algebras over the associated ``adjunction monad''. Despite the words, this is quite elementary mathematics. One can ask the question for any adjunction, one of the most common relations in mathematics, and the answer is usually no.
However, very often the categories involved support a homotopy theory and one can ask whether or not the answer is yes for the associated homotopy categories. Under axioms, the answer is almost yes, almost because one of the axioms is problematic. Under modified axioms, that include all of the operadic kind of iterated and infinite loop space theory, including equivariant and multiplicative, the answer to a question that is simultaneously more and less general is definitely yes, with myriads of calculational applications. There are interesting new examples, one that should be classical but isn't (as far as I know) and one in motivic homotopy theory. I will try to introduce some of the (old) ideas, but will focus on the (new) conceptual framework and maybe some open questions.
If there is time, I will sketch how to elaborate the idea multiplicatively, explaining new analogs of operads, called ``multerads" and ``bioperads". These are as elementary as operads, and their associated monads feed into the general theory and give rise to E-infinity ring spectra and a highly structured version of algebraic K-theory.
